Transmitting apparatus and modulation method thereof

ABSTRACT

A transmitting apparatus is disclosed. The transmitting apparatus includes an encoder to perform channel encoding with respect to bits and generate a codeword, an interleaver to interleave the codeword, and a modulator to map the interleaved codeword onto a non-uniform constellation according to a modulation scheme, and the constellation may include constellation points defined based on various tables according to the modulation scheme.

CROSS-REFERENCE TO THE RELATED APPLICATIONS

This is a continuation of U.S. application Ser. No. 15/725,998 filedOct. 5, 2017, which issued U.S. Pat. No. 10,862,623 on Dec. 8, 2020,which is a continuation of U.S. application Ser. No. 14/615,148 filedFeb. 5, 2015 which issued U.S. Pat. No. 9,813,191 on Nov. 7, 2017; U.S.application Ser. No. 14/615,148 claims priority from Korean PatentApplication No. 10-2015-0017971, filed on Feb. 5, 2015, in the KoreanIntellectual Property Office, and from U.S. Provisional Application No.61/936,029 filed on Feb. 5, 2014 and from U.S. Provisional ApplicationNo. 61/945,868 filed on Feb. 28, 2014 in the United States Patent andTrademark Office, the disclosures of which are incorporated herein intheir entirety by reference.

BACKGROUND Article I. 1. Field

Apparatuses and methods consistent with exemplary embodiments of theinventive concept relate to transmitting and receiving date usingbroadcasting, more particularly, to the design of non-uniformconstellations used in a Bit Interleaved Coded Modulation (BICM) mappingbits at an output of an encoder and interleaver to complexconstellations.

Article II. 2. Description of the Related Art

The current broadcasting systems consistent with the Digital VideoBroadcasting Second Generation Terrestrial (DVB-T2) use a BitInterleaved and Coded Modulation (BICM) chain in order to encode bits tobe transmitted. The BICM chain includes a channel encoder like a LowDensity Parity Check (LDPC) encoder followed by a Bit Interleaver and aQuadrature Amplitude Modulation (QAM) mapper. The role of the QAM mapperis to map different bits output from the channel encoder and interleavedusing the Bit Interleaver to QAM cells. Each cell represents a complexnumber having real and imaginary part. The QAM mapper groups M bits intoone cell. Each cell is translated into a complex number. M, which is thenumber of bits per cell, is equal to 2 for QPSK, 4 for 16QAM, 6 for64QAM, and 8 for 256. It is possible to use a higher QAM size in orderto increase a throughput. For example: 1K QAM is a constellationcontaining 1024 possible points and used to map M=10 bits. The DVB-T2and previous standards use a uniform QAM. The uniform QAM has twoimportant properties: possible points of constellation are rectangular,and spacing between each two successive points is uniform. The uniformQAM is very easy to map and demap.

The QAM is also easy to use since it does not need to be optimised as afunction of the signal to noise ratio (SNR) or the coding rate of thechannel code like the LDPC code. However, the capacity of the uniformQAM leaves a big gap from the theoretical limit, known as the Shannonlimit. The performance in terms of bit error rate (BER) or frame errorrate (FER) may be far from optimal.

SUMMARY

In order to reduce the gap from Shannon limit and provide a betterBER/FER performance, a non-uniform constellation (NUC) is generated byrelaxing the two properties of the uniform QAM, namely: the square shapeand the uniform distance between constellations points.

It is an aim of certain exemplary embodiments of the present inventionto address, solve and/or mitigate, at least partly, at least one of theproblems and/or disadvantages associated with the related art, forexample at least one of the problems and/or disadvantages describedabove. It is an aim of certain exemplary embodiments of the presentinvention to provide at least one advantage over the related art, forexample at least one of the advantages described below.

The present invention is defined in the independent claims. Advantageousfeatures are defined in the dependent claims.

Other aspects, advantages, and salient features of the invention willbecome apparent to those skilled in the art from the following detaileddescription, which, taken in conjunction with the annexed drawings,disclose exemplary embodiments of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and/or other aspects will be more apparent by describingcertain exemplary embodiments with reference to the accompanyingdrawings, in which:

FIG. 1 is a schematic diagram of a first algorithm according to anexemplary embodiment;

FIG. 2 is a flowchart illustrating the operations of the firstalgorithm, according to an exemplary embodiment;

FIG. 3 illustrates the convergence of C_last with respect to one of theparameters as the first algorithm of FIGS. 1 and 2 is performed,according to an exemplary embodiment;

FIG. 4 illustrates a second algorithm according to an exemplaryembodiment for determining an optimal constellation at a given SNR valueS in an AWGN channel;

FIG. 5 illustrates the convergence of the constellation C_best as thesecond algorithm of FIG. 4 is performed, according to an exemplaryembodiment;

FIG. 6 illustrates a third algorithm according to an exemplaryembodiment for determining the optimal constellation at a given SNRvalue S in a Rician fading channel for a desired Rician factor K_rice;

FIG. 7 illustrates a fourth algorithm according to an exemplaryembodiment for determining the optimal constellation at a given SNRvalue S in a Rayleigh fading channel;

FIG. 8 illustrates a fifth algorithm according to an exemplaryembodiment for determining an optimal constellation;

FIG. 9 illustrates a process for obtaining an optimal constellation fora specific system, according to an exemplary embodiment;

FIG. 10 illustrates an exemplary BER versus SNR plot for 64-QAM using aLow-Density Parity-Check, LDPC, coding rate (CR) of 2/3 from DVB-T2 inan AWGN channel, according to an exemplary embodiment;

FIG. 11 illustrates a sixth algorithm according to an exemplaryembodiment for determining an optimal constellation;

FIG. 12 further illustrates the sixth algorithm illustrated in FIG. 11,according to an exemplary embodiment;

FIG. 13 illustrates a process for obtaining the waterfall SNR for acertain channel type according to an exemplary embodiment;

FIG. 14 schematically illustrates a process for obtaining a weightedperformance measure function for an input constellation based ondifferent transmission scenarios according to an exemplary embodiment;

FIG. 15 illustrates a process for obtaining an optimum constellationaccording to an exemplary embodiment;

FIGS. 16A and 16B illustrate alternative schemes for generating acandidate constellation from a previous constellation according toexemplary embodiments;

FIG. 17 illustrates a technique for reducing complexity according to anexemplary embodiment;

FIG. 18 illustrates an apparatus for implementing an algorithm accordingto an exemplary embodiment;

FIGS. 19 to 34 illustrate non uniform constellations according tovarious exemplary embodiments;

FIG. 35 is a block diagram to describe a configuration of a transmittingapparatus according to an exemplary embodiment;

FIG. 36 is a block diagram to describe a configuration of a receivingapparatus according to an exemplary embodiment; and

FIG. 37 is a flowchart to describe a modulation method according to anexemplary embodiment.

DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENTS

Various exemplary embodiments will now be described in greater detailwith reference to the accompanying drawings.

In the following description, same drawing reference numerals are usedfor the same elements even in different drawings. The matters defined inthe description, such as detailed construction and elements, areprovided to assist in a comprehensive understanding of the invention.Thus, it is apparent that the exemplary embodiments can be carried outwithout those specifically defined matters. Also, well-known functionsor constructions are not described in detail since they would obscurethe exemplary embodiments with unnecessary detail.

The following description of the exemplary embodiments with reference tothe accompanying drawings is provided to assist in a comprehensiveunderstanding of the inventive concept, as defined by the claims. Thedescription includes various specific details to assist in thatunderstanding but these are to be regarded as merely exemplary.Accordingly, those of ordinary skill in the art will recognize thatvarious changes and modifications of the embodiments described hereincan be made without departing from the scope of the inventive concept.

The same or similar components may be designated by the same or similarreference numerals, although they may be illustrated in differentdrawings.

Detailed descriptions of techniques, structures, constructions,functions or processes known in the art may be omitted for clarity andconciseness, and to avoid obscuring the subject matter of the exemplaryembodiments.

The terms and words used herein are not limited to the bibliographicalor standard meanings, but, are merely used by the inventors to enable aclear and consistent understanding of the exemplary embodiments.

Throughout the description and claims of this specification, the words“comprise”, “contain” and “include”, and variations thereof, for example“comprising”, “containing” and “including”, means “including but notlimited to”, and is not intended to (and does not) exclude otherfeatures, elements, components, integers, steps, operations, processes,functions, characteristics, and the like.

Throughout the description and claims of this specification, thesingular form, for example “a”, “an” and “the”, encompasses the pluralunless the context otherwise requires. For example, reference to “anobject” includes reference to one or more of such objects.

Throughout the description and claims of this specification, language inthe general form of “X for Y” (where Y is some action, process,function, activity or step and X is some means for carrying out thataction, process, function, activity or step) encompasses means Xadapted, configured or arranged specifically, but not necessarilyexclusively, to do Y.

Features, elements, components, integers, steps, operations, processes,functions, characteristics, and the like, described in conjunction witha particular aspect, embodiment, example or claim of the inventiveconcept are to be understood to be applicable to any other aspect,embodiment, example or claim described herein unless incompatibletherewith.

The exemplary embodiments may be implemented in the form of any suitablemethod, system and/or apparatus for use in digital broadcasting, forexample in the form of a mobile/portable terminal (e.g. mobiletelephone), hand-held device, personal computer, digital televisionand/or digital radio broadcast transmitter and/or receiver apparatus,set-top-box, etc. Any such system and/or apparatus may be compatiblewith any suitable existing or future digital broadcast system and/orstandard, for example one or more of the digital broadcasting systemsand/or standards referred to herein.

A non-uniform constellation (NUC) according to an exemplary embodimentmay be generated or obtained using any suitable method or algorithmincluding steps (or operations) for generating or obtaining such anon-uniform constellation. The non-uniform constellation according tothe embodiment may be generated or obtained by any suitably arrangedapparatus or system including means for generating or obtaining such anon-uniform constellation. The methods or algorithms described hereinmay be implemented in any suitably arranged apparatus or systemincluding means for carrying out the method or algorithm steps.

Certain exemplary embodiments provide an algorithm for obtaining anon-uniform constellation. The non-uniform constellation obtained in thecertain exemplary embodiments may provide a higher capacity than anequivalent uniform constellation (e.g. a uniform constellation of thesame order). Certain exemplary embodiments may obtain an optimisednon-uniform constellation using an algorithm with relatively lowcomplexity and relatively high computational efficiency. For example, analgorithm in certain exemplary embodiments may obtain an optimisednon-uniform constellation much faster than an algorithm using a bruteforce method that searches all (or a high proportion of) possiblecandidate constellations. Certain exemplary embodiments provide analgorithm for obtaining optimised non-uniform constellations suitablefor very high-order constellation (e.g. having more than 1024constellation points).

Various embodiments are described below in which non-uniform (NU)Quadrature Amplitude Modulation (QAM) constellations are obtained.However, the skilled person will appreciate that the inventive conceptis not limited to QAM constellations, but may be applied to other typesof constellation.

As mentioned above, a constellation may be characterised by a number ofparameters, for example specifying spacings between constellationpoints, or specifying the position of each positive real level (completeconstellations may be obtained from these parameters because theconstellations are the same for real and imaginary axes and the same forpositive and negative values). In order to obtain an optimumconstellation, a brute force approach may be taken in which combinationsof values for each of the parameters are searched with a certain stepsize up to a certain maximum value. Each combination of values for eachparameter corresponds to a distinct constellation. The constellationhaving the best performance is selected.

However, in certain exemplary embodiments, the number of parameters maybe reduced by imposing one or more certain geometric and/or symmetryconstraints on the constellations. For example, one constraint may bethat the constellations are symmetric among the four quadrants of theconstellations. In addition, the constellations may be constrained inthat the constellation points are arranged in a QAM type lattice inwhich, within each quadrant, (i) constellation points are arranged inhorizontal and vertical lines, (ii) the number of horizontal lines isthe same as the number of vertical lines, (iii) the same number ofconstellation points are arranged in each horizontal line, and (iv) thesame number of constellation points are arranged in each vertical line.In another example, a constellation may be constrained to be a circularconstellation (e.g. a constellation having circular symmetry).Furthermore, constellations having the same relative arrangement,differing only in size, may be regarded as equivalent. In this case, oneof the parameters may be set to a fixed value. The skilled person willappreciate that the inventive concept is not limited to the aboveexamples, and that one or more additional or alternative constraints maybe used.

In certain exemplary embodiments, a non-uniform QAM (NU-QAM)constellation may have a constellation conforming to one or moregeometric and/or symmetry constraints, for example one or more, or all,of the above constraints, or a rotation and/or scaling thereof. Anon-uniform N-QAM constellation may be a non-uniform QAM constellationincluding N constellation points.

By applying the constraints described above, the number of parametersmay be reduced, for example to 1, 3, 7, 15, 31 and 63 parameter(s) forconstellations including 16, 64, 256, 1024, 4096 and 16384 constellationpoints, respectively. The number of parameters in a reduced set ofparameters may be denoted by b. For example b=1 for 16-QAM (in whichthere are 16 positions that are symmetric on the real/imaginary andpositive/negative axes). Thus there are only 2 points to define. Sincethe total power of the constellation is typically normalized to one,fixing one parameter will fix the other. Thus b=1 for a square 16QAM.

In certain exemplary embodiments, combinations of values for each of theb parameters are searched with a step size d up to a maximum value A.Thus, the number of search iterations is equal to (A/d)^(b).

A first algorithm according to certain exemplary embodiments forobtaining an optimum non-uniform constellation for a given SNR will nowbe described. The algorithm uses an iterative scheme to gradually modifyan initial constellation until the constellation converges. For example,the initial constellation may be a uniform constellation, theconstellation may be modified by changing the values of the parametersbetween iterations, and convergence occurs when the values of all theparameters change by less than a threshold amount between iterations. Anoptimum constellation may be defined as the constellation having thebest performance according to any suitable measure. For example, themeasure may include a coded modulation (CM) capacity or BICM capacity.In the following example a non-uniform 64-QAM constellation is obtained,in which the (reduced) number of variable parameters, b, is equal to 3.

FIG. 1 is a schematic diagram of a first algorithm, according to anexemplary embodiment, and FIG. 2 is a flowchart illustrating operationsof the first algorithm, according to an exemplary embodiment. In thealgorithm, the following variables are used. A parameter C_last denotesa particular constellation, corresponding to a particular set of valuesof the b parameters. A parameter C_last is initialised with a certaininitial constellation, for example a uniform constellation. A parameterSNR denotes a signal-to-noise ratio. The SNR parameter is set to adesired value equal to an SNR for which an optimum constellation isdesired. The parameter C_best denotes a constellation that maximisesperformance, for example maximises the CM capacity or BICM capacity, fora given SNR. The parameter d denotes a first step size used in thealgorithm. The parameter d (or step) is initialised to a suitable valuethat may be determined theoretically and/or experimentally. A parameterMin_Step denotes a minimum allowed value for d, and is set to a fixedvalue.

In operation 201, C_last is initialised to an input constellation. In anext operation 203, step d is initialised to a value Ini_step. Inoperation 205, a set of candidate constellations is obtained. The set ofcandidate constellations includes the constellation C_last and one ormore modified constellations, where each modified constellation isobtained by modifying one or more of the parameter values definingC_last using any suitable scheme. In the illustrated example, the set ofcandidate constellations are created based on C_last and step size d,denoted by function CreateSet(C_Last, d). For example, for eachconstellation point, three derived constellations are generated [C_last,C_last+d, C_last−d]. Specifically, a set of constellations is derivedsuch that the values of the b parameters in C_last are each set to oneof n new values varying around the current parameter value. For example,three new values (n=3) may be used, which include (i) the currentparameter value, (ii) a value d greater than the current parametervalue, and (iii) a value d less than the current parameter value. Forexample, if there are two constellation levels to be defined then thenumber of combinations to be tested are 3×3 (corresponding to threepositions for each level). All combinations of the new parameter valuesare used to generate the set of constellation. Thus, the set ofconstellations includes a total of n^(b) constellations. Although threenew values for each parameter are used in the embodiment describedabove, any suitable number of new values may be used in anotherembodiment. The set of new values may include the old (or current)value, or may not include the old value.

In certain exemplary embodiments, three values of each level are chosenso that the total number of possibilities to be tested is 3^(b), where bis the number of levels (parameters) to be optimised. In the case ofvery high-order constellations, for example above 1K, 3^(b) may be veryhigh. In this case, all the levels may be fixed except one, for whichthree possibilities are tested, C_last, C_last+d and C_last−d untilconvergence is achieved. The same operation may then be repeated for theother levels. The cost of this operation is multiplicative and notexponential (for example, if it is supposed that each level converges inone iteration then the cost will be 3×b instead 3^(b).)

In operation 207, the performance of each constellation in the set ofderived (candidate) constellations is calculated or determined using anysuitable performance measure (e.g. capacity). In operation 209, thecandidate constellation having the best performance (e.g. the candidateconstellation that maximises the capacity) is assigned to C_best. Inoperation 211, it is determined whether C_best differs from C_last bymore than a threshold amount. For example, in the illustrated example,the threshold amount is equal to zero, so that it is determined whetherC_best=C_last. That is, it is determined whether there is any differencebetween constellation C_best and constellation C_last (e.g. within acertain resolution). The difference may be any suitable measure ofdifference, for example including a difference based on geometry (e.g.differences in the locations of the constellation points of theconstellations) and/or a performance measure (e.g. a difference in acertain performance measure between the constellations). If it isdetermined in operation 211 that C_best≠C_last, then in operation 213,C_last takes the value C_best (i.e. so that the value of C_last in thenext iteration is equal to the value of C_best in the current iteration)and the method returns to operation 205 in which a set of candidateconstellations are created based on C_last and step size d,CreateSet(C_Last, d). On the other hand, if it is determined inoperation 211 that C_best=C_last, then, in operation 215, C_last takesthe value C_best and the method moves to operation 217.

In operation 217, it is determined whether d<Min_Step. If it isdetermined in operation 217 that d≥Min_Step then the method moves tooperation 219 in which the step size d is reduced. For example, d isdivided by a certain factor (e.g. 2). Following operation 219, themethod returns to operation 205 in which a set of candidateconstellations are created based on C_last and step size d (i.e.,reduced d), CreateSet(C_Last, d). On the other hand, If it is determinedin operation 217 that d<Min_Step then the value of C_best is saved andthe algorithm ends.

FIG. 3 illustrates the convergence of C_last with respect to one of theparameters as the first algorithm of FIGS. 1 and 2 is performed.Initially, the value of the parameter converges to a certain value. Whenthe value of the parameter has converged within a certain resolution,the step size d is reduced and the value of the parameter convergesfurther, until the step size d has reached the minimum step size.

In the example shown in FIG. 3, for each iteration, three new parametervalues are tried, as represented by the vertical columns of circles. Thebest new parameter for each iteration is indicated in FIG. 3 as a filledcircle. The best parameter value in one iteration is used as the newparameter value for the next iteration. Thus, in the example illustratedin FIG. 3, in which three new parameter values are tried (including thecurrent parameter and parameters an amount d above and below the currentparameter), the filled circle of one iteration corresponds to the middleof the three circles arranged in a column for the next iteration.

In certain exemplary embodiments, operations 217 and 219 of thealgorithm illustrated in FIG. 2 may be omitted so that operations 205,207, 209, 211, 213 and 215 are performed using the initial step size. Inthis case, when it is determined in operation 215 that C_best=C_last,the step size is not reduced, but rather the value of C_best is savedand the algorithm ends. By omitting operations 217 and 219, thealgorithm may potentially complete more quickly. However, in this casethe output constellation C_best may differ from the true optimumconstellation more than the output constellation C_best obtained in thealgorithm illustrated in FIG. 2 where the step size d is decreased. Thismay be seen in FIG. 3, where it can be seen that the best parametervalue in the final iteration lies closer to the optimal value (indicatedby the horizontal line) than the best parameter value at the stage ofconvergence with the initial step size.

The first algorithm described above determines an optimum constellationbased on a certain performance measure (e.g. capacity). In thefollowing, various algorithms for determining an optimum constellationfor a transmission system defined by a set of one or more systemparameter values, where the constellation is optimised for a certaindesired value of a system parameter (e.g. a certain SNR value or certainRicean factor). In these embodiments, a system parameter value is set toan initial value (e.g. a relatively high value) and an optimumconstellation is generated using an algorithm described above (e.g. thealgorithm illustrated in FIG. 2), wherein the performance measure isbased on a defined transmission system having the set system parametervalue. The system parameter value is then reset to a modified value(e.g. by reducing the value by a certain step size) and the algorithm isre-run. The other system parameter values may remain fixed. This processis repeated until the system parameter value reaches a certain desiredvalue.

For example, FIG. 4 illustrates a second algorithm for determining anoptimal constellation at a given SNR value S in an additive whiteGaussian noise (AWGN) channel. In operation 401, the algorithm isinitialised by setting an SNR parameter to a high value N, where N islarge. For example, the initial SNR value may be set to an SNR valueabove which a non-uniform constellation provides no better performancethan an equivalent uniform constellation. This value may be determined,for example, theoretically and/or experimentally. In operation 401, theparameter C_last is also initialised to a certain constellation, forexample a uniform constellation.

In operation 403, the first algorithm described above is run using theinitialised constellation C_last as the input constellation and usingthe initialised SNR ratio. By applying the first algorithm, theconstellation C_last will converge to an optimal constellation C_bestfor a specific input value of SNR. An output of operation 403 is C_bestobtained using the first algorithm. In operation 405 the SNR value isreduced by a certain amount, for example one unit or step size. Inoperation 405, C_last takes the value of C_best (i.e. so that the valueof C_last in the next iteration is equal to the value of C_best in thecurrent iteration). In operation 407 it is determined whether SNR<S. Ifit is determined in operation 407 that SNR≥S, then the method returns tooperation 403, in which the first algorithm is run with new values ofC_last and SNR. On the other hand, if it is determined in operation 407that SNR<S, then the value of C_best is saved and the algorithm ends. Byapplying the second algorithm, the resulting constellation C_best is theoptimal constellation for the desired SNR value S.

FIG. 5 illustrates convergence of the constellation C_best according tothe second algorithm of FIG. 4 is performed. Each of the three curvesrepresents variation in the value of a respective one of the threevariable parameters. The solid constant line represents the fixed valueof a fixed parameter. As shown in FIG. 5, at the start of the secondalgorithm, starting from the right-hand side of FIG. 5, the SNR value ishigh and the constellation is a uniform constellation, as defined by thevalues of the parameters on the right-hand side of FIG. 5, labelled“Initial condition”. At each iteration, an optimal constellation isobtained for the specific SNR value (indicated in FIG. 5 by themarkers). The SNR is then reduced and the optimal constellation isobtained for the new SNR (this process being indicated for one of theparameters by the stepped line in FIG. 5). As shown in FIG. 5, thevalues of the parameters corresponding to the optimal constellation varysmoothly with varying SNR values. The iterations are repeated until theSNR value reaches the desired SNR value S.

By running the second algorithm illustrated in FIG. 4, an optimalconstellation is derived from each of a set of SNR values. Theseconstellations may be stored in association with the corresponding SNRvalues, for example in a look-up table.

FIG. 6 illustrates a third algorithm for determining an optimalconstellation at a given SNR value S in a Rician fading channel for adesired Rician factor K_rice. The Rician channel is given by:

$\begin{matrix}{\sqrt{\frac{K}{K + 1}} + {\sqrt{\frac{1}{K + 1}}h}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

In Equation 1, K is the Rician factor and h is Rayleigh distributed(centred and normalised). Initially, the third algorithm applies thesecond algorithm described above to obtain the optimal constellationC_best at an SNR value S for an AWGN channel, C_best(AWGN). In operation601, parameter C_last is initialised to C_best(AWGN). In operation 601,the Rician factor K is initialised to a high value, which may bedetermined theoretically and/or experimentally. For example, K may beinitialised to a value K_rice+N, where N is large.

In operation 603, the first algorithm described above is run using theinitialised constellation C_last as the input constellation and usingthe initialised Rician factor K to obtain an optimal constellationC_best. In operation 605, the Rician factor K is reduced by a certainamount, for example by one unit. In operation 605, C_last takes thevalue of C_best (i.e. so that the value of C_last in the next iterationis equal to the value of C_best in the current iteration). In operation607 it is determined whether K<K_rice. If it is determined in operation607 that K≥K_rice, then the method returns to operation 603, in whichthe first algorithm is run with new values of C_last and K. On the otherhand, if it is determined in operation 607 that K<K_rice, then the valueof C_best is saved and the algorithm ends. By applying the thirdalgorithm, the resulting constellation C_best is the optimalconstellation for the desired Rician factor K_rice.

FIG. 7 illustrates a fourth algorithm for determining an optimalconstellation at a given SNR value S in a Rayleigh fading channel. ARayleigh fading channel is a special case of Rician fading with theRician factor K=0. Accordingly, the fourth algorithm is the same as thethird algorithm described above, except that K_rice is set to zero.

Table 1 below compares the number of capacity calculation function callsfor obtaining optimal constellations for various constellation sizes(16-QAM, 64-QAM and 256-QAM) using an exhaustive search, a restrictedexhaustive search and an algorithm according to the present embodiment.The values in Table 1 are based on a step size d of 0.0125 and maximumvalue for the parameters of 10. Table 1 also indicates a factordifference between using a restricted exhaustive search and a searchusing an algorithm according to the present embodiment. As can be seen,the algorithm according to the present embodiment is significantly moreefficient, for example by a factor of 1.15×10¹⁰ for 256-QAM.

TABLE 1 Algorithm Restricted according to Gain Exhaustive exhaustiveexemplary versus search search embodiment restricted 16QAM 800 800 21 3864QAM 5.1e9  1.9e8  1701 117577 256QAM  2.1e21 2.5e15 216513 1.15e10

In Table 1, the difference between the exhaustive search and therestrictive exhaustive search is as following. It is assumed in thefollowing that there are 4 levels (parameters) between 0 and 10. In theexhaustive search, each of the 4 parameters is searched over the wholerange [0-10] with a certain granularity. In the case of the restrictedexhaustive search, the range in which each level will fall is fixed. Forexample level 1 (first parameter) will be in the range [0-2.5], level 2in the range [2.5-5], level 3 in the range [5-7.5], level 4 in the range[7.5-10]. In this manner, the number of possibilities is reduced.

FIG. 8 illustrates a fifth algorithm for determining an optimalconstellation. This algorithm corresponds closely to the algorithmillustrated in FIG. 2, but is modified to increase overall efficiency.This algorithm includes an inner loop for operations (operations803-819) corresponding to operations 203-219 of FIG. 2. However,operation 805 for creating a set of candidate constellations is modifiedfrom the corresponding operation 205 of FIG. 2. Specifically, in thealgorithm of FIG. 8, rather than modifying each of the b parameters andtrying all combinations of the new parameters as in the algorithm ofFIG. 2, only one parameter is modified at a time. For example, withinone iteration of the inner loop 803-819, only one parameter (parameteri) is modified to produce a set of candidate constellation. Thecapacities of these constellations are calculated and the bestconstellation is selected, as in FIG. 2.

In the algorithm of FIG. 8, the value of i is varied from 1 to b usingan outer loop (operations 821-825). The algorithm of FIG. 8 isinitialised in operation 801, corresponding to operation 201 of FIG. 2.It can be seen that, by using the algorithm of FIG. 8, rather than thealgorithm of FIG. 2, the total number of candidate constellation tried(i.e. the total number of capacity calculations) is significantlyreduced. However, in simulations, the optimal constellation obtainedusing the algorithm of FIG. 8 is very close to the optimal constellationobtained using the algorithm of FIG. 2, which in turn is very close tothe true optimal constellation obtained using an exhaustive search. Theimprovement in computational efficiency using the algorithms accordingto the above embodiments, when compared to an exhaustive search,increases as the constellation order increases.

As with the algorithm illustrated in FIG. 2, in certain exemplaryembodiments, operations 817 and 819 of the algorithm illustrated in FIG.8 may be omitted.

According to the above embodiments, optimal constellations may beobtained for particular parameters, for example SNR, Rician factor etc.These optimum constellations are obtained independently of anyparticular system implementation, for example independent of aparticular coding scheme. In the following, various embodiments aredescribed for obtaining an optimal constellation for a specifictransmission system.

A transmission system may include a number of processes which may affectthe optimal constellation, for example FEC encoding, bit interleaving,demultiplexing bits to cells, mapping cells to constellations, cellinterleaving, constellation rotation, in-phase and quadrature phase(I/Q) component interleaving, inter-frame convolution and inter-frameblock interleaving, and multiple-inputs-single-output (MISO) precoding.A QAM mapper is used in the Bit Interleaved Coded Modulation (BICM)chain to map bits to symbols. The QAM mapper may use a uniformconstellation to map bits to cells (for example as done in DVB-T2).However, an increase in capacity may be achieved by using a fixednon-uniform constellation. A non-fixed non-uniform constellation (e.g.QAM) may be used to further increase capacity. The BICM capacity dependson the bit to cell mapping used. Optimisations are desirable in the LDPCdesign, the QAM mapping and the mapping of bits to cells.

In certain exemplary embodiments, different constellations are generatedusing a certain step size. A bit error rate (BER), a block error rateand/or a packet error rate corresponding to the constellations areobtained, and the best constellation is selected based on one or more ofthe aforementioned error rates.

In certain exemplary embodiments, the process illustrated in FIG. 9 maybe carried out to obtain an optimal constellation for a specific system.In operation 901, a uniform constellation (e.g. uniform QAM) isselected. In operation 903, BER values for the selected uniformconstellation are obtained over a range of SNR values (e.g. usingsimulation or by obtaining the BER values theoretically orexperimentally). These values may be obtained based on a specificsystem, for example using a particular coding scheme (e.g. LDPC codingwith a certain parity check matrix) with a certain coding rate and acertain bit interleaver and cell interleaver. FIG. 10 illustrates anexemplary plot for 64-QAM using an LDPC coding rate of 2/3 from DVB-T2in an AWGN channel.

In operation 905, an SNR at which the BER falls below a threshold value(e.g. 0.001) is determined. The threshold value may be selected suchthat the resulting SNR falls within a “waterfall zone” of the BER curve(i.e. the zone at which the BER falls relatively rapidly with increasingSNR). The determined SNR value may be denoted S and referred to as a“waterfall” SNR.

Next, an optimal constellation may be obtained for the SNR value Sdetermined in operation 905.

For example, in some exemplary embodiments, in operation 907 a, anoptimal constellation may be selected from optimal constellationsobtained when performing the algorithms described above in relation toFIGS. 1-8 (and stored in a look-up table). Specifically, an optimalconstellation previously determined for the SNR value S may be retrievedfrom the look-up table.

Alternatively, an iterative process may be performed to obtain anoptimal (non-uniform) constellation. Specifically, after operation 905,the process moves to operation 907 b in which the algorithms describedabove in relation to FIGS. 1-8 are used to obtain an optimalconstellation for the SNR value S (or for a value close to S). Afteroperation 907 b, the process returns to operation 903, in which BERvalues are obtained over a range of SNR values. In this iteration, theBER values are obtained for the optimal constellation obtained inoperation 907 b (rather than for the initial uniform constellation as inthe first iteration). In a similar manner as previously described, theSNR value at which the BER falls below a threshold value (using the newset of BER values for the optimal constellation) is determined inoperation 905, and a new optimal constellation for the newly determinedSNR value is obtain in operation 907 b. The previously describedoperations 903, 905, 907 may be repeated a certain number of time (forexample a predetermined number of times). Alternatively, the algorithmmay terminate when the waterfall SNR stops decreasing betweeniterations, and instead starts increasing.

FIGS. 11 and 12 illustrate a sixth algorithm for determining an optimalconstellation. This algorithm corresponds closely to the algorithmillustrated in FIG. 8, but is modified to improve performance. Inparticular, this algorithm introduces a concept of a direction ofconvergence of a parameter value. For example, within the inner loop ofthe algorithm, the direction is initialised to 0. When creating a set ofcandidate constellations, the candidate set depends on the directionparameter. When the best constellation is selected in operation 1109,the direction of convergence of the value of parameter i is obtained.For example, if the parameter value is converging upwards, then thedirection parameter may be set to +1, if the parameter is convergingdownwards, then the direction parameter may be set to −1, and if theparameter does not change, then the direction parameter may be set to 0.As illustrated in FIG. 12, the number of candidate constellations may bereduced when the parameter value is converging upwards or downwards.

As described above, an optimum constellation may be obtained for aparticular system implementation, and/or for certain system parametervalues. For example, an optimum constellation (e.g. a constellation thatoptimises the BICM capacity) may be obtained for a certain propagationchannel type (e.g. AWGN, Rayleigh or Typical Urban, TU6, channel) andfor a certain SNR. However, in some cases, data may be transmitted indifferent scenarios. For example, data may be transmitted throughdifferent types of channels and may be received with different SNRs.Furthermore, it may be desirable or required that a data transmissionsystem uses the same constellation, regardless of the scenario (e.g.channel type or SNR), for example in order to reduce system complexity.In some cases, a transmission system may use a certain constellation formany different scenarios (e.g. channel types and SNRs).

FIGS. 13-16 illustrate an algorithm for obtaining a constellation thatis optimised (e.g. achieves the best capacity) with respect to two ormore different scenarios (e.g. different channel types and/or SNRvalues). The algorithm includes a number of different parts. First, thewaterfall SNR for each channel type (e.g. propagation channel type) isobtained using an algorithm similar to the algorithm illustrated in FIG.9. A weighted performance measure function (e.g. weighted capacity) foran input constellation is defined, based on different scenarios (e.g.different channel types and SNR values). Then, an algorithm similar tothe algorithms illustrated in FIG. 2, 8 or 11 is applied to determine anoptimum constellation, where the performance measure is used based onthe weighted performance measure.

FIG. 13 illustrates a process for obtaining the waterfall SNR for eachchannel type. Each channel type is treated separately in order to obtainits waterfall SNR. In particular, the process illustrated in FIG. 13 isrepeated for each channel type to obtain a respective waterfall SNR forthat channel type. The process illustrated in FIG. 13 operates insubstantially the same manner as the algorithm illustrated in FIG. 9,and therefore a detailed description will be omitted for conciseness.However, rather than outputting an optimal constellation, as in thealgorithm illustrated in FIG. 9, the process illustrated in FIG. 13instead outputs the waterfall SNR determined in the final iteration ofthe process. The process illustrated in FIG. 13 (including BERsimulation and capacity optimisation operations) is performed based on acertain channel type, and the output waterfall SNR is determined as thewaterfall SNR associated with that channel type.

FIG. 14 schematically illustrates a process for obtaining a weightedperformance measure function for an input constellation based ondifferent transmission scenarios. In this example, the weightedperformance measure is a weighted capacity, and the different scenariosinclude different channel types and associated waterfall SNR values. Asillustrated in FIG. 14, a candidate constellation is provided as aninput. For each channel type and associated waterfall SNR, the BICMcapacity for the input constellation based on the channel type andwaterfall SNR is obtained. Each obtained BICM capacity is thenmultiplied by a respective weight and the weighted BICM capacities areadded together to obtain an output weighted average BICM capacity. Theweights may be selected according to any suitable criteria. For example,a relatively common or important channel type may be associated with arelatively large weight.

FIG. 15 illustrates a process for obtaining an optimum constellation.The process illustrated in FIG. 15 operates in substantially the samemanner as the algorithm illustrated in FIG. 2, 8 or 11, and therefore adetailed description will be omitted for conciseness. However, whendetermining the performance of a candidate constellation in the processillustrated in FIG. 15, the performance is determined based on theweighted performance measure described above in relation to FIG. 14.

In the process illustrated in FIG. 15, in some situation, a certainconstellation may achieve the best performance with respect to theweighted performance measure, even though the performance of thatconstellation with respect to the BICM capacity based on an individualchannel and SNR may be relatively low. In certain exemplary embodiments,to ensure that a constellation obtained using the algorithm is able toachieve at least a certain level of performance for one or more, or all,transmission scenarios, an additional criterion may be applied whentesting each candidate constellation to obtain the constellation C_best.Specifically, any candidate constellation that does not achieve at leasta threshold performance with respect to one or more certain individualscenarios, or all scenarios, is ignored and cannot be selected asC_best, even if that constellation achieves the best performance withrespect to the weighted performance measure.

In the process illustrated in FIG. 15, a set of candidate constellationsmay be derived using any suitable method, for example the methoddescribed above in relation to FIG. 9 based on a step size d. FIGS. 16Aand 16B illustrate alternative schemes for generating a candidateconstellation from a previous constellation, C_last, that may be used incertain exemplary embodiments. In FIGS. 16A and 16B, the open circlesrepresent the constellation points of a previous constellation, C_last.For each constellation point of the previous constellation, a respectiveset of N modified constellation points are defined, as indicated inFIGS. 16A and 16B as filled circles. Each set of modified constellationpoints forms a pattern of constellation points located relatively closeto the respective constellation point of the previous constellation.

For example, as illustrated in FIG. 16A, each set of modifiedconstellation points may form a square or rectangular lattice of N=8constellation points surrounding a respective constellation point of theprevious constellation. The lattice spacing is equal to d.Alternatively, as illustrated in FIG. 16B, each set of modifiedconstellation points may form a ring of N=8 constellation pointssurrounding a respective constellation point of the previousconstellation. The radius of the ring is equal to d.

A candidate constellation may be obtained by selecting, for eachconstellation point in the previous constellation, either theconstellation point of the previous constellation itself or one of theconstellation points of a respective set of modified constellationpoints.

In the examples described above, a weighted performance measure isdefined based on different transmission scenarios. For example, in thecase illustrated in FIG. 14, each transmission scenario includes adifferent channel type and an associated waterfall SNR value.Accordingly, a constellation optimised for a range of channel types andassociated SNR values may be obtained. In an alternative embodiment, anoptimal constellation may be obtained for different transmissionscenarios, in the case where each transmission scenario includes thesame channel type, but involves different SNR values (e.g. a set of SNRvalues S1, S1+d, S1+2d, S1+3d, . . . , S2, where d is a step size). Thatis, an optimal constellation may be obtained for a fixed channel typethat is intended to be used over a range of SNR values. In this case,the algorithm described above in relation to FIGS. 13-16 may be used,except that when determining the weighted performance measure asillustrated in FIG. 14, instead of determining individual BICMcapacities based on respective channel types and associated waterfallSNR values, the individual BICM capacities are determined based on thefixed channel type and respective SNR values S1, S1+d, S1+2d, S1+3d, . .. , S2.

In the algorithms described above, a technique may be applied to reducethe overall complexity. In particular, when a set of candidateconstellations is generated and the performance of the candidateconstellations are tested, those candidate constellations that have beenpreviously tested (i.e. in one or more previous iteration) are notre-tested. That is, in a current iteration, only those candidateconstellations that have not been tested in previous iterations aretested.

For example, as described above, a first set of candidateconstellations, A, is generated in an iteration, and the best performingcandidate constellation, a (a∈A), is selected from this set. In a nextiteration, a second set of candidate constellations, B, is generatedbased on the previously selected constellation a (a∈B). In this nextiteration, the best performing candidate constellation b (b∈B) from setB needs to be determined.

Typically, there will be at least some overlap between the two sets ofcandidate constellations A and B, such that one or more candidateconstellations belong to both sets A and B (i.e. A∩B≠Ø), includingconstellation a. Since it is known that constellation a has the bestperformance of all the constellations in set A, then it is also knownthat constellation a has the best performance of all the constellationsbelonging to the overlap between sets A and B (i.e. A∩B).

Accordingly, when testing the constellations in set B to determine thebest performing constellation, b, it is not necessary to re-test thoseconstellations belonging to the overlap between sets A and B (i.e. it isnot necessary to re-test those constellations in the set AnB). Instead,rather than testing all constellations in set B, only thoseconstellations belonging to the smaller set of constellations B*,including constellations belonging to set B but excluding anyconstellations that also belong to set A (i.e. B*=B−A) are tested. Then,the best performing constellation from the set formed from the union ofB* and the previous best performing constellation, a (i.e. the bestperforming constellation from the set B*∪{a}) is selected as the bestperforming constellation, b, of set B.

An example of the above principle in relation to the example shown inFIG. 16A is illustrated in FIG. 17. In the example of FIG. 17, atiteration i, it was found that the constellation point indicated as ablack circle is the best performing constellation. At iteration i+1,there is no need to test the common subset (including the white circlesand the black circle), because it was already tested before and gave aninferior performance. That is, at iteration i+1, only the dark greycircles need to be tested. Accordingly, in the illustrated example, areduction in complexity of 44% (=4/9) is achieved.

FIG. 18 illustrates an apparatus for implementing an algorithm accordingto one or more of the embodiments described above. The apparatus isconfigured for generating a non-uniform constellation. The apparatusincludes a block for performing a first process. The block forperforming the first process includes: a block for obtaining a firstconstellation defined by one or more parameter values; and a block forgenerating a second constellation based on the first constellation usinga second process. The block for generating the second constellationbased on the first constellation using the second process includes: ablock for obtaining a set of candidate constellations, wherein the setof candidate constellations includes the first constellation and one ormore modified constellations, wherein each modified constellation isobtained by modifying the parameter values defining the firstconstellation; a block for determining performance of each candidateconstellation according to a predetermined performance measure; and ablock for selecting the candidate constellation having the bestperformance as the second constellation. The block for performing thefirst process further includes a block for determining a differencebetween the first constellation and the second constellation; and ablock for, if the second constellation differs from the firstconstellation by more than a threshold amount, causing the block forperforming the first process to repeat the first process using thesecond constellation generated in a current iteration of the firstprocess as the first constellation in a next iteration.

The skilled person will appreciate that the functions of any two or moreblocks illustrated in FIG. 18 may be performed by a single block, andthat the functions of any block illustrated in FIG. 18 may be performedby two or more blocks. A block may be implemented in any suitable form,for example hardware, software, firmware, or any suitable combination ofhardware, software and firmware.

A constellation obtained by a method according to the above exemplaryembodiments may be used in a digital broadcasting system to transmitdata from a transmitter side to a receiver side. In certain exemplaryembodiments, the system includes a transmitter arranged to obtain data(e.g. a data stream), perform any required encoding and/or otherprocessing of the data, modulate a signal using the data according to amodulation scheme corresponding to the constellation, and transmit themodulated signal. The system further includes a receiver configured toreceive a modulated signal, demodulate the signal according to ademodulation scheme corresponding to the constellation (or a similar orcorresponding constellation), and perform any necessary decoding and/orother processing to recover the original data. Certain exemplaryembodiments may include a transmitter side apparatus only, a receiverside apparatus only, or a system including both a transmitter sideapparatus and a receiver side apparatus.

In case of non-uniform constellations, it is possible to designconstellations by relaxing only one constraint, i.e. by keeping aconstellations square but changing a distance between constellationpoints. This form of non-uniform constellations (NUCs) can be referredto as one-dimensional (1D) NUCs. 1D NUCs can be described by the levelsat which the constellations occur in the real positive part. The otherpoints can be deduced by using the symmetry of the four quadrants aswell as the real and imaginary symmetry. 1D-NUCs are simple to decodebecause of independence of the real and imaginary part. Two (2) pulseamplitude modulator (PAM) demappers can be used to decode 1D-NUCs.

It is possible to design a different type of NUC by relaxing bothconstraints: the square shape and the uniform distance betweenconstellation points. Optimal constellations will have a tendency tolook like a circular constellation. This type of NUC can be referred toas 2D-NUC. The 2D-NUC has a higher capacity than the 1D-NUC and a betterBER/FER performance. However, performance of 2D-NUC comes at the expenseof a more complex receiver demapper. Since the real and imaginary axesare not symmetrical, a 2D demapper is needed in order to decode a 2D-NUCconstellation. In the case of 2D-NUC, a complete set of points need tobe specified. It is possible to specify only the points belonging to thefirst quadrant and deduce the other points by supposing that theconstellation is symmetrical.

Optimizing 1D and 2D NUCs depend on an SNR at which a capacity needs tobe optimized. In the case of a BICM chain, an SNR can be selected to bea waterfall SNR of a BER/FER curve. The BER/FER waterfall can be definedas an SNR at which a BER curve falls below a certain level, for example10e-6. The waterfall SNR depends on a coding rate of an LDPCencoder/decoder. As the code rate increases, the waterfall SNRincreases. For this reason, a different NUC is associated with each LDPCencoder coding rate. The waterfall SNR also increases with a QAMconstellation size (M). This is because the a receiver needs a higherSNR to decode a higher QAM constellation. Thus, a constellations sizeand a coding rate define the waterfall SNR. The waterfall SNR is used tooptimize the constellations. Here, the coding rates include: 2/15, 3/15and 4/15. The NUC sizes are: 16QAM, 64QAM, 256QAM and 1K QAM. For thefirst three QAM sizes only 2D constellations are proposed. For 1K QAMboth 1D and 2D constellations are proposed.

Hereinafter, an example of constellation points of a constellationobtained by applying the algorithms described above by coding rates willbe described.

In the following exemplary embodiment, a restriction is added to theprocess of determining capacity according to SNR with respect to theexisting NUC designing method.

When SNR is given, it is general to calculate the maximum transmissioncapacity which can be transmitted with error-free. In other words, whencalculating capacity by setting SNR with respect to BER/FER waterfall,the SNR indicates an area where a bit error or a frame error occurs, butactual capacity indicates transmission capacity under error-freecircumstances, and thus, there may be a contradiction.

Therefore, in the present disclosure, when calculating capacity withrespect to the SNR, a correction factor is added.

For example, when SNR1 is decided with respect to CH1 in FIG. 1, if thecapacity value under error-free state is C1, corrected C1 value, that isC1′ is defined as shown below.C1′=C1×(1−H(P _(b)))  [Equation 2]

Here, P_(b) indicates a BER value which determines a waterfall area, andfunction H(x) indicates binary entropy function,H(x)=−x×log₂(x)−(1−x)×log₂(1−x).

In this case, the reason why the value (1−H(P_(b))) which is equal to orsmaller than 1 is multiplied to the existing capacity value as indicatedin Equation 2 is as shown below.

With respect to a case where a bit error occurs as much as theprobability P_(b) in a transmission channel, if it is assumed that a biterror as much as P_(b) has been occurred before transmitting sourceinformation, and error does not occur while source information is beingtransmitted, there is no difference in terms of a bit error in the finaltransmitting/receiving end. As described above, when the probability oferror occurrence regarding a channel is considered a loss of sourceinformation, there is an effect as if lossy compression is applied asmuch as H(P_(b)) compared to given data, by the rate distortion theoryof Shannon. That is, in conclusion, it can be considered that dataamount which can be transmitted through firstly given channel, that iscapacity, will be reduced compared to the channel (1−H(P_(b))) withoutan error or loss.

When the same value is applied to all the channels which consider P_(b)of FIG. 14, the same factor(1−H(P_(b))) is multiplied, and thus, thesame factor (1−H(P_(b))) is multiplied to the value of weightedcapacity. Accordingly, there is no difference in the constellationpoints of the optimized NUC. However, target BER may be different foreach channel, and therefore, the order of size of weighted capacity canbe different. Accordingly, the constellation points of optimized NUC canbe different. For example, an AWGN channel is applied to a fixed devicein general, and thus, very low BER is required. Therefore, whencalculating capacity, BER=1e−8 may be considered. As Rayleigh channel islargely considered for a mobile channel which experiences fading, BERhigher than that of AWGN channel is required. Thus, BER=1e−6 can beconsidered. As described above, if different BER requirements are setfor respective channels, values of factor for final capacity becomedifferent, and therefore, this value may be different from the NUCconstellation points which are obtained in consideration of capacityonly without reflecting BER.

To be specific, Table 2 indicates values of constellation points of anormalized 2D NU 16-QAM constellation (2D 16NUC) which is obtained byapplying the algorithms described above using respective coding rates2/15, 3/15 and 4/15 for a single SNR value.

TABLE 2 Coding Rate w/Shape 2/15 3/15 4/15 w0 0.707316 + 0.707473i 0.709789 + 0.710216i  1.10696 + 0.580648i w1 0.707333 + 0.707367i 0.705044 + 0.710131i 0.581442 + 1.10703i w2 0.706049 + 0.707738i0.709276 + 0.70389i  0.552546 + 0.362564i w3 0.706475 + 0.707102i 0.704489 + 0.703976i 0.362944 + 0.55265i w4 −0.707316 + 0.707473i −0.709789 + 0.710216i  −1.10696 + 0.580648i w5 −0.707333 + 0.707367i −0.705044 + 0.710131i −0.581442 + 1.10703i  w6 −0.706049 + 0.707738i −0.709276 + 0.70389i  −0.552546 + 0.362564i w7 −0.706475 + 0.707102i −0.704489 + 0.703976i −0.362944 + 0.55265i  w8 0.707316 − 0.707473i 0.709789 − 0.710216i  1.10696 − 0.580648i w9 0.707333 − 0.707367i 0.705044 − 0.710131i 0.581442 − 1.10703i w10 0.706049 − 0.707738i0.709276 − 0.70389i  0.552546 − 0.362564i w11 0.706475 − 0.707102i 0.704489 − 0.703976i 0.362944 − 0.55265i w12 −0.707316 − 0.707473i −0.709789 − 0.710216i  −1.10696 − 0.580648i w13 −0.707333 − 0.707367i −0.705044 − 0.710131i −0.581442 − 1.10703i  w14 −0.706049 − 0.707738i −0.709276 − 0.70389i  −0.552546 − 0.362564i w15 −0.706475 − 0.707102i −0.704489 − 0.703976i −0.362944 − 0.55265i 

In this case, the constellation points of the 2D NU 16-QAM constellationfor respective coding rates of 2/15, 3/15, and 4/15 are indicated inFIGS. 19-21.

Table 3 indicates values of constellation points of a normalized 2D NU64-QAM constellation (2D 64NUC) which is obtained by applying thealgorithms described above using respective coding rates 2/15, 3/15 and4/15 for a single SNR value.

TABLE 3 Coding Rate w/Shape 2/15 3/15 4/15 w0  0.647424 + 0.983083i0.547191 + 1.15913i 0.500831 + 1.21361i w1  0.643837 + 0.982885i 0.54734 + 1.15734i 0.499379 + 1.21941i w2 0.647063 + 0.97668i0.546743 + 1.15987i 0.531316 + 1.17151i w3  0.644354 + 0.976223i0.547937 + 1.15853i 0.529865 + 1.17877i w4  0.983862 + 0.647524i 1.15778 + 0.547787i    1.2107 + 0.503734i w5  0.977782 + 0.647422i 1.15763 + 0.547489i  1.22087 + 0.500831i w6  0.983498 + 0.643369i 1.15913 + 0.547489i  1.17151 + 0.529865i w7  0.977678 + 0.643265i 1.15913 + 0.547489i  1.18022 + 0.526961i w8  0.465916 + 0.639338i 0.316261 + 0.507211i  0.274368 + 0.476152i w9 0.464253 + 0.63861i 0.316261 + 0.507211i  0.272917 + 0.476152i w10  0.466124 + 0.635284i 0.316261 + 0.507211i  0.277272 + 0.479056i w11  0.463941 + 0.634972i 0.316261 + 0.507211i  0.277272 + 0.479056i w12 0.637769 + 0.46706i 0.508702 + 0.316261i  0.476152 + 0.272917i w13  0.635171 + 0.467267i 0.508702 + 0.316261i  0.476152 + 0.272917i w14  0.638497 + 0.465604i 0.508702 + 0.316261i  0.479056 + 0.277272i w15  0.635275 + 0.465293i 0.508702 + 0.316261i 0.479056 + 0.27582i w16 −0.647424 + 0.983083i−0.547191 + 1.15913i  −0.500831 + 1.21361i  w17 −0.643837 + 0.982885i−0.54734 + 1.15734i −0.499379 + 1.21941i  w18 −0.647063 + 0.97668i −0.546743 + 1.15987i  −0.531316 + 1.17151i  w19 −0.644354 + 0.976223i−0.547937 + 1.15853i  −0.529865 + 1.17877i  w20 −0.983862 + 0.647524i −1.15778 + 0.547787i  −1.2107 + 0.503734i w21 −0.977782 + 0.647422i −1.15763 + 0.547489i  −1.22087 + 0.500831i w22 −0.983498 + 0.643369i −1.15913 + 0.547489i  −1.17151 + 0.529865i w23 −0.977678 + 0.643265i −1.15913 + 0.547489i  −1.18022 + 0.526961i w24 −0.465916 + 0.639338i−0.316261 + 0.507211i −0.274368 + 0.476152i w25 −0.464253 + 0.63861i −0.316261 + 0.507211i −0.272917 + 0.476152i w26 −0.466124 + 0.635284i−0.316261 + 0.507211i −0.277272 + 0.479056i w27 −0.463941 + 0.634972i−0.316261 + 0.507211i −0.277272 + 0.479056i w28 −0.637769 + 0.46706i −0.508702 + 0.316261i −0.476152 + 0.272917i w29 −0.635171 + 0.467267i−0.508702 + 0.316261i −0.476152 + 0.272917i w30 −0.638497 + 0.465604i−0.508702 + 0.316261i −0.479056 + 0.277272i w31 −0.635275 + 0.465293i−0.508702 + 0.316261i −0.479056 + 0.27582i  w32  0.647424 − 0.983083i0.547191 − 1.15913i 0.500831 − 1.21361i w33  0.643837 − 0.982885i 0.54734 − 1.15734i 0.499379 − 1.21941i w34 0.647063 − 0.97668i 0.546743− 1.15987i 0.531316 − 1.17151i w35  0.644354 − 0.976223i 0.547937 −1.15853i 0.529865 − 1.17877i w36  0.983862 − 0.647524i  1.15778 −0.547787i   1.2107 − 0.503734i w37  0.977782 − 0.647422i  1.15763 −0.547489i  1.22087 − 0.500831i w38  0.983498 − 0.643369i  1.15913 −0.547489i  1.17151 − 0.529865i w39  0.977678 − 0.643265i  1.15913 −0.547489i  1.18022 − 0.526961i w40  0.465916 − 0.639338i  0.316261 −0.507211i  0.274368 − 0.476152i w41 0.464253 − 0.63861i  0.316261 −0.507211i  0.272917 − 0.476152i w42  0.466124 − 0.635284i  0.316261 −0.507211i  0.277272 − 0.479056i w43  0.463941 − 0.634972i  0.316261 −0.507211i  0.277272 − 0.479056i w44 0.637769 − 0.46706i  0.508702 −0.316261i  0.476152 − 0.272917i w45  0.635171 − 0.467267i  0.508702 −0.316261i  0.476152 − 0.272917i w46  0.638497 − 0.465604i  0.508702 −0.316261i  0.479056 − 0.277272i w47  0.635275 − 0.465293i  0.508702 −0.316261i 0.479056 − 0.27582i w48 −0.647424 − 0.983083i −0.547191 −1.15913i  −0.500831 − 1.21361i  w49 −0.643837 − 0.982885i −0.54734 −1.15734i −0.499379 − 1.21941i  w50 −0.647063 − 0.97668i  −0.546743 −1.15987i  −0.531316 − 1.17151i  w51 −0.644354 − 0.976223i −0.547937 −1.15853i  −0.529865 − 1.17877i  w52 −0.983862 − 0.647524i  −1.15778 −0.547787i  −1.2107 − 0.503734i w53 −0.977782 − 0.647422i  −1.15763 −0.547489i  −1.22087 − 0.500831i w54 −0.983498 − 0.643369i  −1.15913 −0.547489i  −1.17151 − 0.529865i w55 −0.977678 − 0.643265i  −1.15913 −0.547489i  −1.18022 − 0.526961i w56 −0.465916 − 0.639338i −0.316261 −0.507211i −0.274368 − 0.476152i w57 −0.464253 − 0.63861i  −0.316261 −0.507211i −0.272917 − 0.476152i w58 −0.466124 − 0.635284i −0.316261 −0.507211i −0.277272 − 0.479056i w59 −0.463941 − 0.634972i −0.316261 −0.507211i −0.277272 − 0.479056i w60 −0.637769 − 0.46706i  −0.508702 −0.316261i −0.476152 − 0.272917i w61 −0.635171 − 0.467267i −0.508702 −0.316261i −0.476152 − 0.272917i w62 −0.638497 − 0.465604i −0.508702 −0.316261i −0.479056 − 0.277272i w63 −0.635275 − 0.465293i −0.508702 −0.316261i −0.479056 − 0.27582i 

In this case, the constellation points of the 2D NU 64-QAM constellationfor respective coding rates of 2/15, 3/15 and 4/15 are indicated inFIGS. 22-24.

Table 4 indicates values of constellation points of a normalized 2D NU256-QAM constellation (2D 256NUC) which is obtained by applying thealgorithms described above using respective coding rates 2/15, 3/15 and4/15 for a single SNR value.

TABLE 4 Coding Rate w/Shape 2/15 3/15 4/15 w0 0.555322 + 1.12624i0.522922 + 1.18101i 0.297463 + 1.05643i w1  0.56728 + 1.13359i0.538432 + 1.16254i 0.586177 + 0.96165i w2 0.559308 + 1.12037i0.514797 + 1.1943i  0.290901 + 1.06955i w3 0.563576 + 1.13205i0.528831 + 1.1751i   0.579615 + 0.968941i w4 0.552505 + 1.12494i0.498548 + 1.22015i 0.295276 + 1.33567i w5 0.563704 + 1.13202i0.511105 + 1.19725i  0.74876 + 1.23651i w6 0.559846 + 1.11805i0.488947 + 1.23566i 0.300379 + 1.51137i w7 0.565901 + 1.1274i 0.504457 + 1.21129i 0.815106 + 1.3816i  w8 0.557904 + 1.1381i 0.522183 + 1.18174i 0.300379 + 1.05351i w9 0.561681 + 1.14706i0.536955 + 1.16402i  0.584718 + 0.963109i w10 0.559343 + 1.13456i 0.51332 + 1.19504i 0.293088 + 1.06591i w11 0.567186 + 1.14299i0.530308 + 1.1751i   0.582531 + 0.966754i w12 0.553299 + 1.13551i0.497071 + 1.22163i 0.295276 + 1.3189i  w13 0.563201 + 1.14214i0.512582 + 1.19947i 0.746573 + 1.21683i w14 0.556748 + 1.13251i0.488208 + 1.23714i 0.296005 + 1.46544i w15 0.564101 + 1.13628i0.504457 + 1.21277i 0.829688 + 1.35389i w16  1.13089 + 0.559658i 1.17953 + 0.525138i  1.06372 + 0.296005i w17  1.14052 + 0.565983i 1.16254 + 0.538432i  0.96165 + 0.581073i w18  1.13482 + 0.558817i 1.19135 + 0.51332i    1.0732 + 0.293088i w19  1.14905 + 0.563844i 1.17436 + 0.529569i  0.968212 + 0.581802i w20  1.12445 + 0.561493i 1.22089 + 0.499287i  1.36191 + 0.29965i w21  1.13332 + 0.562675i 1.20021 + 0.514797i  1.22485 + 0.754593i w22  1.12837 + 0.557788i 1.23418 + 0.488208i  1.54272 + 0.310586i w23  1.14364 + 0.563576i 1.21424 + 0.505196i  1.39691 + 0.852289i w24  1.11959 + 0.562018i 1.18027 + 0.522922i  1.06153 + 0.294546i w25  1.13468 + 0.566507i 1.16402 + 0.53991i  0.963109 + 0.581802i w26  1.13792 + 0.56113i 1.19208 + 0.51332i  1.07101 + 0.292359i w27  1.14396 + 0.563832i 1.17584 + 0.530308i  0.967483 + 0.582531i w28  1.12214 + 0.559424i 1.22089 + 0.497071i  1.32546 + 0.297463i w29  1.13182 + 0.568564i 1.20243 + 0.514797i  1.19787 + 0.749489i w30  1.13018 + 0.56189i 1.23492 + 0.488947i  1.45596 + 0.304024i w31  1.13855 + 0.566158i 1.21498 + 0.504457i  1.32692 + 0.841353i w32  0.33943 + 0.53805i 0.274017 + 0.477129i  0.249344 + 0.558472i w33 0.339746 + 0.53604i 0.276233 + 0.480084i  0.296005 + 0.534412i w34  0.33867 + 0.532417i 0.273278 + 0.475652i  0.244969 + 0.541703i w35  0.340014 + 0.533493i 0.274756 + 0.477868i  0.287256 + 0.519102i w36  0.337373 + 0.530582i 0.270324 + 0.474175i  0.20487 + 0.392243i w37  0.340506 + 0.534252i 0.27254 + 0.476391i  0.217264 + 0.380577i w38  0.337911 + 0.532417i 0.269585 + 0.472698i  0.199038 + 0.375474i w39  0.340014 + 0.531704i 0.271801 + 0.474914i  0.210703 + 0.364538i w40  0.339652 + 0.536975i 0.274017 + 0.477868i 0.249344 + 0.55993i w41 0.339968 + 0.53832i 0.275494 + 0.479345i  0.297463 + 0.535141i w42  0.338133 + 0.534696i 0.27254 + 0.475652i 0.244969 + 0.54389i w43  0.338179 + 0.534743i 0.274756 + 0.477868i  0.288714 + 0.521289i w44  0.337864 + 0.534158i 0.271063 + 0.473436i  0.205599 + 0.393701i w45  0.338939 + 0.533177i 0.27254 + 0.476391i  0.218723 + 0.382035i w46  0.34019 + 0.534696i 0.269585 + 0.471959i  0.199767 + 0.376203i w47  0.338449 + 0.533983i 0.271063 + 0.474175i  0.212161 + 0.366725i w48 0.534965 + 0.33943i 0.477129 + 0.274017i  0.560659 + 0.248615i w49  0.536309 + 0.339746i 0.478607 + 0.276233i  0.538058 + 0.296005i w50  0.534205 + 0.338939i0.476391 + 0.27254i  0.543 89 + 0.24424i w51  0.538366 + 0.337957i 0.477129 + 0.274756i  0.522018 + 0.286527i w52  0.532907 + 0.336344i 0.473436 + 0.270324i 0.390784 + 0.20487i w53  0.532955 + 0.338717i0.475652 + 0.27254i  0.381306 + 0.217264i w54  0.53112 + 0.338939i 0.473436 + 0.269585i  0.374016 + 0.199767i w55  0.533223 + 0.337957i 0.474175 + 0.271063i  0.365267 + 0.209974i w56  0.531342 + 0.339652i 0.477129 + 0.274017i  0.564304 + 0.248615i w57  0.532417 + 0.339968i 0.477868 + 0.276233i  0.540974 + 0.296734i w58 0.533936 + 0.34019i0.476391 + 0.27254i  0.547536 + 0.243511i w59  0.53604 + 0.340506i 0.477129 + 0.274756i  0.525663 + 0.287985i w60  0.528525 + 0.339652i 0.474175 + 0.270324i 0.393701 + 0.20487i w61  0.531658 + 0.337911i0.474914 + 0.27254i  0.384952 + 0.218723i w62  0.531879 + 0.338133i 0.473436 + 0.269585i  0.376203 + 0.199767i w63  0.532686 + 0.339477i 0.474914 + 0.271063i  0.368912 + 0.211432i w64 −0.555322 + 1.12624i −0.522922 + 1.18101i  −0.297463 + 1.05643i  w65 −0.56728 + 1.13359i−0.538432 + 1.16254i  −0.586177 + 0.96165i  w66 −0.559308 + 1.12037i −0.514797 + 1.1943i  −0.290901 + 1.06955i  w67 −0.563576 + 1.13205i −0.528831 + 1.1751i  −0.579615 + 0.968941i w68 −0.552505 + 1.12494i −0.498548 + 1.22015i  −0.295276 + 1.33567i  w69 −0.563704 + 1.13202i −0.511105 + 1.19725i  −0.74876 + 1.23651i w70 −0.559846 + 1.11805i −0.488947 + 1.23566i  −0.300379 + 1.51137i  w71 −0.565901 + 1.1274i −0.504457 + 1.21129i  −0.815106 + 1.3816i  w72 −0.557904 + 1.1381i −0.522183 + 1.18174i  −0.300379 + 1.05351i  w73 −0.561681 + 1.14706i −0.536955 + 1.16402i  −0.584718 + 0.963109i w74 −0.559343 + 1.13456i −0.51332 + 1.19504i −0.293088 + 1.06591i  w75 −0.567186 + 1.14299i −0.530308 + 1.1751i  −0.582531 + 0.966754i w76 −0.553299 + 1.13551i −0.497071 + 1.22163i  −0.295276 + 1.3189i  w77 −0.563201 + 1.14214i −0.512582 + 1.19947i  −0.746573 + 1.21683i  w78 −0.556748 + 1.13251i −0.488208 + 1.23714i  −0.296005 + 1.46544i  w79 −0.564101 + 1.13628i −0.504457 + 1.21277i  −0.829688 + 1.35389i  w80  −1.13089 + 0.559658i −1.17953 + 0.525138i  −1.06372 + 0.296005i w81  −1.14052 + 0.565983i −1.16254 + 0.538432i  −0.96165 + 0.581073i w82  −1.13482 + 0.558817i−1.19135 + 0.51332i  −1.0732 + 0.293088i w83  −1.14905 + 0.563844i −1.17436 + 0.529569i −0.968212 + 0.581802i w84  −1.12445 + 0.561493i −1.22089 + 0.499287i −1.36191 + 0.29965i w85  −1.13332 + 0.562675i −1.20021 + 0.514797i  −1.22485 + 0.754593i w86  −1.12837 + 0.557788i −1.23418 + 0.488208i  −1.54272 + 0.310586i w87  −1.14364 + 0.563576i −1.21424 + 0.505196i  −1.39691 + 0.852289i w88  −1.11959 + 0.562018i −1.18027 + 0.522922i  −1.06153 + 0.294546i w89  −1.13468 + 0.566507i−1.16402 + 0.53991i −0.963109 + 0.581802i w90 −1.13792 + 0.56113i−1.19208 + 0.51332i  −1.07101 + 0.292359i w91  −1.14396 + 0.563832i −1.17584 + 0.530308i −0.967483 + 0.582531i w92  −1.12214 + 0.559424i −1.22089 + 0.497071i  −1.32546 + 0.297463i w93  −1.13182 + 0.568564i −1.20243 + 0.514797i  −1.19787 + 0.749489i w94 −1.13018 + 0.56189i −1.23492 + 0.488947i  −1.45596 + 0.304024i w95  −1.13855 + 0.566158i −1.21498 + 0.504457i  −1.32692 + 0.841353i w96 −0.33943 + 0.53805i−0.274017 + 0.477129i −0.249344 + 0.558472i w97 −0.339746 + 0.53604i −0.276233 + 0.480084i −0.296005 + 0.534412i w98  −0.33867 + 0.532417i−0.273278 + 0.475652i −0.244969 + 0.541703i w99 −0.340014 + 0.533493i−0.274756 + 0.477868i −0.287256 + 0.519102i w100 −0.337373 + 0.530582i−0.270324 + 0.474175i  −0.20487 + 0.392243i w101 −0.340506 + 0.534252i −0.27254 + 0.476391i −0.217264 + 0.380577i w102 −0.337911 + 0.532417i−0.269585 + 0.472698i −0.199038 + 0.375474i w103 −0.340014 + 0.531704i−0.271801 + 0.474914i −0.210703 + 0.364538i w104 −0.339652 + 0.536975i−0.274017 + 0.477868i −0.249344 + 0.55993i  w105 −0.339968 + 0.53832i −0.275494 + 0.479345i −0.297463 + 0.535141i w106 −0.338133 + 0.534696i −0.27254 + 0.475652i −0.244969 + 0.54389i  w107 −0.338179 + 0.534743i−0.274756 + 0.477868i −0.288714 + 0.521289i w108 −0.337864 + 0.534158i−0.271063 + 0.473436i −0.205599 + 0.393701i w109 −0.338939 + 0.533177i −0.27254 + 0.476391i −0.218723 + 0.382035i w110  −0.34019 + 0.534696i−0.269585 + 0.471959i −0.199767 + 0.376203i w111 −0.338449 + 0.533983i−0.271063 + 0.474175i −0.212161 + 0.366725i w112 −0.534965 + 0.33943i −0.477129 + 0.274017i −0.560659 + 0.248615i w113 −0.536309 + 0.339746i−0.478607 + 0.276233i −0.538058 + 0.296005i w114 −0.534205 + 0.338939i−0.476391 + 0.27254i  −0.54389 + 0.24424i w115 −0.538366 + 0.337957i−0.477129 + 0.274756i −0.522018 + 0.286527i w116 −0.532907 + 0.336344i−0.473436 + 0.270324i −0.390784 + 0.20487i  w117 −0.532955 + 0.338717i−0.475652 + 0.27254i  −0.381306 + 0.217264i w118  −0.53112 + 0.338939i−0.473436 + 0.269585i −0.374016 + 0.199767i w119 −0.533223 + 0.337957i−0.474175 + 0.271063i −0.365267 + 0.209974i w120 −0.531342 + 0.339652i−0.477129 + 0.274017i −0.564304 + 0.248615i w121 −0.532417 + 0.339968i−0.477868 + 0.276233i −0.540974 + 0.296734i w122 −0.533936 + 0.34019i −0.476391 + 0.27254i  −0.547536 + 0.243511i w123  −0.53604 + 0.340506i−0.477129 + 0.274756i −0.525663 + 0.287985i w124 −0.528525 + 0.339652i−0.474175 + 0.270324i −0.393701 + 0.20487i  w125 −0.531658 + 0.337911i−0.474914 + 0.27254i  −0.384952 + 0.218723i w126 −0.531879 + 0.338133i−0.473436 + 0.269585i −0.376203 + 0.199767i w127 −0.532686 + 0.339477i−0.474914 + 0.271063i −0.368912 + 0.211432i w128 0.555322 − 1.12624i0.522922 − 1.18101i 0.297463 − 1.05643i w129  0.56728 − 1.13359i0.538432 − 1.16254i 0.586177 − 0.96165i w130 0.559308 − 1.12037i0.514797 − 1.1943i  0.290901 − 1.06955i w131 0.563576 − 1.13205i0.528831 − 1.1751i   0.579615 − 0.968941i w132 0.552505 − 1.12494i0.498548 − 1.22015i 0.295276 − 1.33567i w133 0.563704 − 1.13202i0.511105 − 1.19725i  0.74876 − 1.23651i w134 0.559846 − 1.11805i0.488947 − 1.23566i 0.300379 − 1.51137i w135 0.565901 − 1.1274i 0.504457 − 1.21129i 0.815106 − 1.3816i  w136 0.557904 − 1.1381i 0.522183 − 1.18174i 0.300379 − 1.05351i w137 0.561681 − 1.14706i0.536955 − 1.16402i  0.584718 − 0.963109i w138 0.559343 − 1.13456i 0.51332 − 1.19504i 0.293088 − 1.06591i w139 0.567186 − 1.14299i0.530308 − 1.1751i   0.582531 − 0.966754i w140 0.553299 − 1.13551i0.497071 − 1.22163i 0.295276 − 1.3189i  w141 0.563201 − 1.14214i0.512582 − 1.19947i 0.746573 − 1.21683i w142 0.556748 − 1.13251i0.488208 − 1.23714i 0.296005 − 1.46544i w143 0.564101 − 1.13628i0.504457 − 1.21277i 0.829688 − 1.35389i w144  1.13089 − 0.559658i 1.17953 − 0.525138i  1.06372 − 0.296005i w145  1.14052 − 0.565983i 1.16254 − 0.538432i  0.96165 − 0.581073i w146  1.13482 − 0.558817i 1.19135 − 0.51332i   1.0732 − 0.293088i w147  1.14905 − 0.563844i 1.17436 − 0.529569i  0.968212 − 0.581802i w148  1.12445 − 0.561493i 1.22089 − 0.499287i  1.36191 − 0.29965i w149  1.13332 − 0.562675i 1.20021 − 0.514797i  1.22485 − 0.754593i w150  1.12837 − 0.557788i 1.23418 − 0.488208i  1.54272 − 0.310586i w151  1.14364 − 0.563576i 1.21424 − 0.505196i  1.39691 − 0.852289i w152  1.11959 − 0.562018i 1.18027 − 0.522922i  1.06153 − 0.294546i w153  1.13468 − 0.566507i 1.16402 − 0.53991i  0.963109 − 0.581802i w154  1.13792 − 0.56113i 1.19208 − 0.51332i  1.07101 − 0.292359i w155  1.14396 − 0.563832i 1.17584 − 0.530308i  0.967483 − 0.582531i w156  1.12214 − 0.559424i 1.22089 − 0.497071i  1.32546 − 0.297463i w157  1.13182 − 0.568564i 1.20243 − 0.514797i  1.19787 − 0.749489i w158  1.13018 − 0.56189i 1.23492 − 0.488947i  1.45596 − 0.304024i w159  1.13855 − 0.566158i 1.21498 − 0.504457i  1.32692 − 0.841353i w160  0.33943 − 0.53805i 0.274017 − 0.477129i  0.249344 − 0.558472i w161 0.339746 − 0.53604i 0.276233 − 0.480084i  0.296005 − 0.534412i w162  0.33867 − 0.532417i 0.273278 − 0.475652i  0.244969 − 0.541703i w163  0.340014 − 0.533493i 0.274756 − 0.477868i  0.287256 − 0.519102i w164  0.337373 − 0.530582i 0.270324 − 0.474175i  0.20487 − 0.392243i w165  0.340506 − 0.534252i 0.27254 − 0.476391i  0.217264 − 0.380577i w166  0.337911 − 0.532417i 0.269585 − 0.472698i  0.199038 − 0.375474i w167  0.340014 − 0.531704i 0.271801 − 0.474914i  0.210703 − 0.364538i w168  0.339652 − 0.536975i 0.274017 − 0.477868i 0.249344 − 0.55993i w169 0.339968 − 0.53832i 0.275494 − 0.479345i  0.297463 − 0.535141i w170  0.338133 − 0.534696i 0.27254 − 0.475652i 0.244969 − 0.54389i w171  0.338179 − 0.534743i 0.274756 − 0.477868i  0.288714 − 0.521289i w172  0.337864 − 0.534158i 0.271063 − 0.473436i  0.205599 − 0.393701i w173  0.338939 − 0.533177i 0.27254 − 0.476391i  0.218723 − 0.382035i w174  0.34019 − 0.534696i 0.269585 − 0.471959i  0.199767 − 0.376203i w175  0.338449 − 0.533983i 0.271063 − 0.474175i  0.212161 − 0.366725i w176 0.534965 − 0.33943i 0.477129 − 0.274017i  0.560659 − 0.248615i w177  0.536309 − 0.339746i 0.478607 − 0.276233i  0.538058 − 0.296005i w178  0.534205 − 0.338939i0.476391 − 0.27254i  0.54389 − 0.24424i w179  0.538366 − 0.337957i 0.477129 − 0.274756i  0.522018 − 0.286527i w180  0.532907 − 0.336344i 0.473436 − 0.270324i 0.390784 − 0.20487i w181  0.532955 − 0.338717i0.475652 − 0.27254i  0.381306 − 0.217264i w182  0.53112 − 0.338939i 0.473436 − 0.269585i  0.374016 − 0.199767i w183  0.533223 − 0.337957i 0.474175 − 0.271063i  0.365267 − 0.209974i w184  0.531342 − 0.339652i 0.477129 − 0.274017i  0.564304 − 0.248615i w185  0.532417 − 0.339968i 0.477868 − 0.276233i  0.540974 − 0.296734i w186 0.533936 − 0.34019i0.476391 − 0.27254i  0.547536 − 0.243511i w187  0.53604 − 0.340506i 0.477129 − 0.274756i  0.525663 − 0.287985i w188  0.528525 − 0.339652i 0.474175 − 0.270324i 0.393701 − 0.20487i w189  0.531658 − 0.337911i0.474914 − 0.27254i  0.384952 − 0.218723i w190  0.531879 − 0.338133i 0.473436 − 0.269585i  0.376203 − 0.199767i w191  0.532686 − 0.339477i 0.474914 − 0.271063i  0.368912 − 0.211432i w192 −0.555322 − 1.12624i −0.522922 − 1.18101i  −0.297463 − 1.05643i  w193 −0.56728 − 1.13359i−0.538432 − 1.16254i  −0.586177 − 0.96165i  w194 −0.559308 − 1.12037i −0.514797 − 1.1943i  −0.290901 − 1.06955i  w195 −0.563576 − 1.13205i −0.528831 − 1.1751i  −0.579615 − 0.968941i w196 −0.552505 − 1.12494i −0.498548 − 1.22015i  −0.295276 − 1.33567i  w197 −0.563704 − 1.13202i −0.511105 − 1.19725i  −0.74876 − 1.23651i w198 −0.559846 − 1.11805i −0.488947 − 1.23566i  −0.300379 − 1.51137i  w199 −0.565901 − 1.1274i −0.504457 − 1.21129i  −0.815106 − 1.3816i  w200 −0.557904 − 1.1381i −0.522183 − 1.18174i  −0.300379 − 1.05351i  w201 −0.561681 − 1.14706i −0.536955 − 1.16402i  −0.584718 − 0.963109i w202 −0.559343 − 1.13456i −0.51332 − 1.19504i −0.293088 − 1.06591i  w203 −0.567186 − 1.14299i −0.530308 − 1.1751i  −0.582531 − 0.966754i w204 −0.553299 − 1.13551i −0.497071 − 1.22163i  −0.295276 − 1.3189i  w205 −0.563201 − 1.14214i −0.512582 − 1.19947i  −0.746573 − 1.21683i  w206 −0.556748 − 1.13251i −0.488208 − 1.23714i  −0.296005 − 1.46544i  w207 −0.564101 − 1.13628i −0.504457 − 1.21277i  −0.829688 − 1.35389i  w208  −1.13089 − 0.559658i −1.17953 − 0.525138i  −1.06372 − 0.296005i w209  −1.14052 − 0.565983i −1.16254 − 0.538432i  −0.96165 − 0.581073i w210  −1.13482 − 0.558817i−1.19135 − 0.51332i  −1.0732 − 0.293088i w211  −1.14905 − 0.563844i −1.17436 − 0.529569i −0.968212 − 0.581802i w212  −1.12445 − 0.561493i −1.22089 − 0.499287i −1.36191 − 0.29965i w213  −1.13332 − 0.562675i −1.20021 − 0.514797i  −1.22485 − 0.754593i w214  −1.12837 − 0.557788i −1.23418 − 0.488208i  −1.54272 − 0.310586i w215  −1.14364 − 0.563576i −1.21424 − 0.505196i  −1.39691 − 0.852289i w216  −1.11959 − 0.562018i −1.18027 − 0.522922i  −1.06153 − 0.294546i w217  −1.13468 − 0.566507i−1.16402 − 0.53991i −0.963109 − 0.581802i w218 −1.13792 − 0.56113i−1.19208 − 0.51332i  −1.07101 − 0.292359i w219  −1.14396 − 0.563832i −1.17584 − 0.530308i −0.967483 − 0.582531i w220  −1.12214 − 0.559424i −1.22089 − 0.497071i  −1.32546 − 0.297463i w221  −1.13182 − 0.568564i −1.20243 − 0.514797i  −1.19787 − 0.749489i w222 −1.13018 − 0.56189i −1.23492 − 0.488947i  −1.45596 − 0.304024i w223  −1.13855 − 0.566158i −1.21498 − 0.504457i  −1.32692 − 0.841353i w224 −0.33943 − 0.53805i−0.274017 − 0.477129i −0.249344 − 0.558472i w225 −0.339746 − 0.53604i −0.276233 − 0.480084i −0.296005 − 0.534412i w226  −0.33867 − 0.532417i−0.273278 − 0.475652i −0.244969 − 0.541703i w227 −0.340014 − 0.533493i−0.274756 − 0.477868i −0.287256 − 0.519102i w228 −0.337373 − 0.530582i−0.270324 − 0.474175i  −0.20487 − 0.392243i w229 −0.340506 − 0.534252i −0.27254 − 0.476391i −0.217264 − 0.380577i w230 −0.337911 − 0.532417i−0.269585 − 0.472698i −0.199038 − 0.375474i w231 −0.340014 − 0.531704i−0.271801 − 0.474914i −0.210703 − 0.364538i w232 −0.339652 − 0.536975i−0.274017 − 0.477868i −0.249344 − 0.55993i  w233 −0.339968 − 0.53832i −0.275494 − 0.479345i −0.297463 − 0.535141i w234 −0.338133 − 0.534696i −0.27254 − 0.475652i −0.244969 − 0.54389i  w235 −0.338179 − 0.534743i−0.274756 − 0.477868i −0.288714 − 0.521289i w236 −0.337864 − 0.534158i−0.271063 − 0.473436i −0.205599 − 0.393701i w237 −0.338939 − 0.533177i −0.27254 − 0.476391i −0.218723 − 0.382035i w238  −0.34019 − 0.534696i−0.269585 − 0.471959i −0.199767 − 0.376203i w239 −0.338449 − 0.533983i−0.271063 − 0.474175i −0.212161 − 0.366725i w240 −0.534965 − 0.33943i −0.477129 − 0.274017i −0.560659 − 0.248615i w241 −0.536309 − 0.339746i−0.478607 − 0.276233i −0.538058 − 0.296005i w242 −0.534205 − 0.338939i−0.476391 − 0.27254i  −0.54389 − 0.24424i w243 −0.538366 − 0.337957i−0.477129 − 0.274756i −0.522018 − 0.286527i w244 −0.532907 − 0.336344i−0.473436 − 0.270324i −0.390784 − 0.20487i  w245 −0.532955 − 0.338717i−0.475652 − 0.27254i  −0.381306 − 0.217264i w246  −0.53112 − 0.338939i−0.473436 − 0.269585i −0.374016 − 0.199767i w247 −0.533223 − 0.337957i−0.474175 − 0.271063i −0.365267 − 0.209974i w248 −0.531342 − 0.339652i−0.477129 − 0.274017i −0.564304 − 0.248615i w249 −0.532417 − 0.339968i−0.477868 − 0.276233i −0.540974 − 0.296734i w250 −0.533936 − 0.34019i −0.476391 − 0.27254i  −0.547536 − 0.243511i w251  −0.53604 − 0.340506i−0.477129 − 0.274756i −0.525663 − 0.287985i w252 −0.528525 − 0.339652i−0.474175 − 0.270324i −0.393701 − 0.20487i  w253 −0.531658 − 0.337911i−0.474914 − 0.27254i  −0.384952 − 0.218723i w254 −0.531879 − 0.338133i−0.473436 − 0.269585i −0.376203 − 0.199767i w255 −0.532686 − 0.339477i−0.474914 − 0.271063i −0.368912 − 0.211432i

In this case, the constellation points of the 2D NU 256-QAMconstellation for the respective coding rates of 2/15, 3/15, and 4/15are illustrated in FIGS. 25-27.

Meanwhile, according to Tables 2-4, when values of constellation pointsare determined in one quadrant, values of constellation points in otherquadrants may be deduced by symmetry. For example, for eachconstellation point A in the top-right quadrant, correspondingconstellation points may be present in three different quadrants(bottom-right, bottom-left and top-left) respectively, and they can beindicated as A*, −A*, and −A. Here, * indicates complex conjugation.

Table 5 indicates values of constellation points of a normalized 1D NU1024-QAM constellation (1D 1024NUC) which is obtained by applying thealgorithms described above using respective coding rates 2/15, 3/15 and4/15 for a single SNR value.

TABLE 5 Coding Rate Level 2/15 3/15 4/15 1 1 1 1 2 1.0009888421.073113208 1.008229665 3 1.001438042 1.153930818 1.075789474 41.001284289 1.073899371 1.065263158 5 1.002909247 1.1534591191.588516746 6 1.00312028 1.238050314 1.608803828 7 1.001525471.153930818 1.502200957 8 1.0015315 1.073742138 1.483062201 92.824341802 3.982232704 3.724784689 10 2.839150319 3.5044025163.826411483 11 2.868001604 3.278616352 3.897799043 12 2.8550260633.504559748 3.807272727 13 2.848080048 3.972484277 5.023923445 142.861384198 3.499371069 5.023349282 15 2.831923931 3.9707547175.88937799 16 2.818221832 6.002830189 7.67138756

In this case, the constellation points of the 1D NU 1K QAM constellationfor the respective coding rates 2/15, 3/15 and 4/15 may be illustratedin FIGS. 28-30.

For example, FIG. 28 illustrates an exemplary 1D NU 1024-QAMconstellation obtained by applying the algorithms described above usingthe coding rate of 2/15.

According to FIG. 28, a complete set of constellation points areindicated in a constellation diagram on the right-hand side of thedrawing. Values of the constellation points of the top-right quadrantare indicated on the top-left side of the drawing.

In the case of the 1D NU 1K QAM constellation, rather than giving thevalues of the constellation points explicitly, a set of levels of theconstellation points are given instead, from which actual values of theconstellation points may be deduced. To be specific, given a set of mlevels A=[A₁, A₂, . . . , A_(m)], a set of m² constellation point valuesC+Di may be deduced. Herein, C and D each may include a value selectedfrom a level set A. A complete set of constellation points in thetop-right quadrant may be obtained by considering all possible pairs ofvalues C and D. According to FIG. 28, values of constellation points inthe remaining three quadrants may be similarly deduced by symmetry. Asan example, according to Table 5, when the coding rate is 2/15, A={1,1.000988842, . . . , 2.818221832}, and a group of C+Di corresponding toa constellation point set of the first quadrant has 256 elements such as{1+i, 1+1.000988842×i, 1.000988842+i, . . . ,2.818221832+2.818221832×i}, the complete set of 1D NU 1024-QAMconstellation points may be obtained by indicating an arbitrary elementa in the group, as a*, −a* and −a. Here, * indicates complexconjugation.

Table 6 indicates values of constellation points of a normalized 1D NU4096-QAM constellation (1D 4096NUC) which is obtained by applying thealgorithms described above using respective coding rates 2/15, 3/15,4/15 and 5/15 for a single SNR value.

TABLE 6 Coding Rate Level 2/15 3/15 4/15 5/15 1 1 1 1 1 2 1.020833333 10.973180077 1 3 1.041666667 1.057142857 0.965517241 1 4 1.0208333331.057142857 0.980842912 1 5 1.041666667 1.057142857 0.988505747 1 61.0625 1.057142857 0.957854406 1 7 1.041666667 1.028571429 0.988505747 18 1.020833333 1 1 1 9 1.041666667 1.371428571 2.233716475 2.863636364 101.0625 1.428571429 2.233716475 2.863636364 11 1.0625 1.4857142862.210727969 2.772727273 12 1.0625 1.457142857 2.229885057 2.772727273 131.041666667 1.457142857 2.245210728 2.772727273 14 1.0625 1.4857142862.22605364 2.772727273 15 1.041666667 1.371428571 2.2145593872.863636364 16 1.020833333 1.371428571 2.237547893 2.863636364 17 3.43753.542857143 4.275862069 4.863636364 18 3.041666667 3.6285714294.356321839 4.863636364 19 2.791666667 3.657142857 4.7088122615.545454545 20 3 3.6 4.62835249 5.590909091 21 2.916666667 3.6285714294.624521073 5.545454545 22 2.75 3.685714286 4.724137931 5.545454545 232.958333333 3.657142857 4.360153257 4.863636364 24 3.25 3.6 4.2873563224.863636364 25 4.125 6.4 8.98467433 11.18181818 26 3.3541666675.314285714 7.911877395 9.772727273 27 2.979166667 4.8 6.708812261 8 283.291666667 4.971428571 6.536398467 7.545454545 29 3.5208333334.857142857 6.425287356 7.5 30 3.083333333 4.714285714 6.6091954027.909090909 31 3.604166667 5.228571429 7.747126437 9.454545455 32 5.18758.057142857 10.8697318 13.18181818

In this case, the constellation points of the 1D NU 4096-QAMconstellation for the respective coding rates 2/15, 3/15, 4/15 and 5/15may be illustrated in FIGS. 31-34.

In the case of the 1D NU 4K QAM constellation, rather than giving thevalues of the constellation points explicitly, a set of levels of theconstellation points are given instead, from which actual values of theconstellation points may be deduced. To be specific, given a set of mlevels A=[A₁, A₂, . . . , A_(m)], and a set of m² constellation pointvalues C+Di may be deduced. Herein, C and D each may include a valueselected from a level set A. A complete set of constellation points inthe top-right quadrant may be obtained by considering all possible pairsof values C and D. That is, values of constellation points in theremaining three quadrants may be similarly deduced by symmetry. As anexample, according to Table 6, when the coding rate is 2/15, A={1,1.020833333, . . . , 5.1875}, and a group of C+Di corresponding to aconstellation point set of the first quadrant has 256 elements such as{1+i, 1+1.020833333×i, 1.020833333+i, . . . , 5.1875+5.1875×i}, thecomplete set of 1D NU 4096-QAM constellation points may be obtained byindicating an arbitrary element a in the group, as a*, −a* and −a.Here, * indicates complex conjugation.

As described above, in the 1D-NUC of Tables 5 and 6, the constellationcan be described by the levels at which the constellations occur in thereal positive part. The constellation points can be deduced by using thereal/imaginary symmetry but also the symmetry of the four quadrants.

Meanwhile, the inventive concept is not limited to the constellationsdefined in Tables 2-6.

For example, when different sizes of normalization and rounding-off areapplied for the values of constellation points defined in Tables 2-6,the values can be indicated as in Tables 7-11. In this case, theconstellation defined in Tables 7-11 can be an exemplary embodiment.

Tables 7-11 illustrate the set of constellation points for one quadrantonly, but it is obvious to obtain a complete set of constellation pointsby indicating the constellation point a in the one quadrant, as a*, −a*and −a. Here, * indicates complex conjugation.

To be specific, Table 7 indicates the values of constellation points of2D NU 16-QAM constellation (2D 16NUC) which is obtained by applyingnormalization and rounding-off of the values of constellation pointsdefined in Table 2.

TABLE 7 w/Shape NUC_16_2/15 NUC_16_3/15 NUC_16_4/15 w0 0.7073 + 0.7075i0.7098 + 0.7102i 1.1070 + 0.5806i w1 0.7073 + 0.7074i 0.7050 + 0.7101i0.5814 + 1.1070i w2 0.7060 + 0.7077i 0.7093 + 0.7039i 0.5525 + 0.3626iw3 0.7065 + 0.7071i 0.7045 + 0.7040i 0.3629 + 0.5527i

In this case, the values of constellation points of other quadrants canbe determined by symmetry.

Table 8 indicates values of constellation points of a 2D NU 64-QAMconstellation (2D 64NUC) which is obtained by normalization and applyingrounding-off of the values of constellation points defined in Table 3.

TABLE 8 w/Shape NUC_64_2/15 NUC_64_3/15 NUC_64_4/15 w0 0.6474 + 0.9831i0.5472 + 1.1591i 0.5008 + 1.2136i w1 0.6438 + 0.9829i 0.5473 + 1.1573i0.4994 + 1.2194i w2 0.6471 + 0.9767i 0.5467 + 1.1599i 0.5313 + 1.1715iw3 0.6444 + 0.9762i 0.5479 + 1.1585i 0.5299 + 1.1788i w4 0.9839 +0.6475i 1.1578 + 0.5478i 1.2107 + 0.5037i w5 0.9778 + 0.6474i 1.1576 +0.5475i 1.2209 + 0.5008i w6 0.9835 + 0.6434i 1.1591 + 0.5475i 1.1715 +0.5299i w7 0.9777 + 0.6433i 1.1591 + 0.5475i 1.1802 + 0.5270i w80.4659 + 0.6393i 0.3163 + 0.5072i 0.2744 + 0.4762i w9 0.4643 + 0.6386i0.3163 + 0.5072i 0.2729 + 0.4762i w10 0.4661 + 0.6353i 0.3163 + 0.5072i0.2773 + 0.4791i w11 0.4639 + 0.6350i 0.3163 + 0.5072i 0.2773 + 0.4791iw12 0.6378 + 0.4671i 0.5087 + 0.3163i 0.4762 + 0.2729i w13 0.6352 +0.4673i 0.5087 + 0.3163i 0.4762 + 0.2729i w14 0.6385 + 0.4656i 0.5087 +0.3163i 0.4791 + 0.2773i w15 0.6353 + 0.4653i 0.5087 + 0.3163i 0.4791 +0.2758i

In this case, the values of constellation points of other quadrants canbe determined by symmetry.

Table 9 indicates values of constellation points of a 2D NU 256-QAMconstellation (2D 256NUC) which is obtained by applying normalizationand rounding-off of the values of constellation points defined in Table4.

TABLE 9 w/Shape NUC_256_2/15 NUC_256_3/15 NUC_256_4/15 w0 0.5553 +1.1262i 0.5229 + 1.1810i 0.2975 + 1.0564i w1 0.5673 + 1.1336i 0.5384 +1.1625i 0.5862 + 0.9617i w2 0.5593 + 1.1204i 0.5148 + 1.1943i 0.2909 +1.0696i w3 0.5636 + 1.1321i 0.5288 + 1.1751i 0.5796 + 0.9689i w40.5525 + 1.1249i 0.4985 + 1.2202i 0.2953 + 1.3357i w5 0.5637 + 1.1320i0.5111 + 1.1973i 0.7488 + 1.2365i w6 0.5598 + 1.1181i 0.4889 + 1.2357i0.3004 + 1.5114i w7 0.5659 + 1.1274i 0.5045 + 1.2113i 0.8151 + 1.3816iw8 0.5579 + 1.1381i 0.5222 + 1.1817i 0.3004 + 1.0535i w9 0.5617 +1.1471i 0.5370 + 1.1640i 0.5847 + 0.9631i w10 0.5593 + 1.1346i 0.5133 +1.1950i 0.2931 + 1.0659i w11 0.5672 + 1.1430i 0.5303 + 1.1751i 0.5825 +0.9668i w12 0.5533 + 1.1355i 0.4971 + 1.2216i 0.2953 + 1.3189i w130.5632 + 1.1421i 0.5126 + 1.1995i 0.7466 + 1.2168i w14 0.5567 + 1.1325i0.4882 + 1.2371i 0.2960 + 1.4654i w15 0.5641 + 1.1363i 0.5045 + 1.2128i0.8297 + 1.3539i w16 1.1309 + 0.5597i 1.1795 + 0.5251i 1.0637 + 0.2960iw17 1.1405 + 0.5660i 1.1625 + 0.5384i 0.9617 + 0.5811i w18 1.1348 +0.5588i 1.1914 + 0.5133i 1.0732 + 0.2931i w19 1.1491 + 0.5638i 1.1744 +0.5296i 0.9682 + 0.5818i w20 1.1245 + 0.5615i 1.2209 + 0.4993i 1.3619 +0.2997i w21 1.1333 + 0.5627i 1.2002 + 0.5148i 1.2249 + 0.7546i w221.1284 + 0.5578i 1.2342 + 0.4882i 1.5427 + 0.3106i w23 1.1436 + 0.5636i1.2142 + 0.5052i 1.3969 + 0.8523i w24 1.1196 + 0.5620i 1.1803 + 0.5229i1.0615 + 0.2945i w25 1.1347 + 0.5665i 1.1640 + 0.5399i 0.9631 + 0.5818iw26 1.1379 + 0.5611i 1.1921 + 0.5133i 1.0710 + 0.2924i w27 1.1440 +0.5638i 1.1758 + 0.5303i 0.9675 + 0.5825i w28 1.1221 + 0.5594i 1.2209 +0.4971i 1.3255 + 0.2975i w29 1.1318 + 0.5686i 1.2024 + 0.5148i 1.1979 +0.7495i w30 1.1302 + 0.5619i 1.2349 + 0.4889i 1.4560 + 0.3040i w311.1386 + 0.5662i 1.2150 + 0.5045i 1.3269 + 0.8414i w32 0.3394 + 0.5381i0.2740 + 0.4771i 0.2493 + 0.5585i w33 0.3397 + 0.5360i 0.2762 + 0.4801i0.2960 + 0.5344i w34 0.3387 + 0.5324i 0.2733 + 0.4757i 0.2450 + 0.5417iw35 0.3400 + 0.5335i 0.2748 + 0.4779i 0.2873 + 0.5191i w36 0.3374 +0.5306i 0.2703 + 0.4742i 0.2049 + 0.3922i w37 0.3405 + 0.5343i 0.2725 +0.4764i 0.2173 + 0.3806i w38 0.3379 + 0.5324i 0.2696 + 0.4727i 0.1990 +0.3755i w39 0.3400 + 0.5317i 0.2718 + 0.4749i 0.2107 + 0.3645i w400.3397 + 0.5370i 0.2740 + 0.4779i 0.2493 + 0.5599i w41 0.3400 + 0.5383i0.2755 + 0.4793i 0.2975 + 0.5351i w42 0.3381 + 0.5347i 0.2725 + 0.4757i0.2450 + 0.5439i w43 0.3382 + 0.5347i 0.2748 + 0.4779i 0.2887 + 0.5213iw44 0.3379 + 0.5342i 0.2711 + 0.4734i 0.2056 + 0.3937i w45 0.3389 +0.5332i 0.2725 + 0.4764i 0.2187 + 0.3820i w46 0.3402 + 0.5347i 0.2696 +0.4720i 0.1998 + 0.3762i w47 0.3384 + 0.5340i 0.2711 + 0.4742i 0.2122 +0.3667i w48 0.5350 + 0.3394i 0.4771 + 0.2740i 0.5607 + 0.2486i w490.5363 + 0.3397i 0.4786 + 0.2762i 0.5381 + 0.2960i w50 0.5342 + 0.3389i0.4764 + 0.2725i 0.5439 + 0.2442i w51 0.5384 + 0.3380i 0.4771 + 0.2748i0.5220 + 0.2865i w52 0.5329 + 0.3363i 0.4734 + 0.2703i 0.3908 + 0.2049iw53 0.5330 + 0.3387i 0.4757 + 0.2725i 0.3813 + 0.2173i w54 0.5311 +0.3389i 0.4734 + 0.2696i 0.3740 + 0.1998i w55 0.5332 + 0.3380i 0.4742 +0.2711i 0.3653 + 0.2100i w56 0.5313 + 0.3397i 0.4771 + 0.2740i 0.5643 +0.2486i w57 0.5324 + 0.3400i 0.4779 + 0.2762i 0.5410 + 0.2967i w580.5339 + 0.3402i 0.4764 + 0.2725i 0.5475 + 0.2435i w59 0.5360 + 0.3405i0.4771 + 0.2748i 0.5257 + 0.2880i w60 0.5285 + 0.3397i 0.4742 + 0.2703i0.3937 + 0.2049i w61 0.5317 + 0.3379i 0.4749 + 0.2725i 0.3850 + 0.2187iw62 0.5319 + 0.3381i 0.4734 + 0.2696i 0.3762 + 0.1998i w63 0.5327 +0.3395i 0.4749 + 0.2711i 0.3689 + 0.2114i

In this case, the values of constellation points of other quadrants canbe determined by symmetry.

Table 10 indicates values of constellation points of a 1D NU 1024-QAMconstellation (1D 1024NUC) which is obtained by applying normalizationand rounding-off of the values of constellation points defined in Table5.

TABLE 10 u/Shape NUC_1k_2/15 NUC_1k_3/15 NUC_1k_4/15 u0 0.3317 0.23820.1924 u1 0.3321 0.2556 0.1940 u2 0.3322 0.2749 0.2070 u3 0.3321 0.25580.2050 u4 0.3327 0.2748 0.3056 u5 0.3328 0.2949 0.3096 u6 0.3322 0.27490.2890 u7 0.3322 0.2558 0.2854 u8 0.9369 0.9486 0.7167 u9 0.9418 0.83480.7362 u10 0.9514 0.7810 0.7500 u11 0.9471 0.8348 0.7326 u12 0.94480.9463 0.9667 u13 0.9492 0.8336 0.9665 u14 0.9394 0.9459 1.1332 u150.9349 1.4299 1.4761

Table 11 indicates values of constellation points of a 1D NU 4096-QAMconstellation (1D 4096NUC) which is obtained by applying normalizationand rounding-off of the values of constellation points defined in Table6.

TABLE 11 u/Shape NUC_4k_2/15 NUC_4k_3/15 NUC_4k_4/15 NUC_4k_5/15 u00.2826 0.2038 0.1508 0.1257 u1 0.2885 0.2038 0.1468 0.1257 u2 0.29440.2155 0.1456 0.1257 u3 0.2885 0.2155 0.1479 0.1257 u4 0.2944 0.21550.1491 0.1257 u5 0.3003 0.2155 0.1444 0.1257 u6 0.2944 0.2097 0.14910.1257 u7 0.2885 0.2038 0.1508 0.1257 u8 0.2944 0.2796 0.3368 0.3599 u90.3003 0.2912 0.3368 0.3599 u10 0.3003 0.3029 0.3334 0.3484 u11 0.30030.2970 0.3363 0.3484 u12 0.2944 0.2970 0.3386 0.3484 u13 0.3003 0.30290.3357 0.3484 u14 0.2944 0.2796 0.3340 0.3599 u15 0.2885 0.2796 0.33740.3599 u16 0.9714 0.7222 0.6448 0.6112 u17 0.8596 0.7397 0.6569 0.6112u18 0.7889 0.7455 0.7101 0.6969 u19 0.8478 0.7339 0.6979 0.7026 u200.8242 0.7397 0.6974 0.6969 u21 0.7771 0.7513 0.7124 0.6969 u22 0.83600.7455 0.6575 0.6112 u23 0.9184 0.7339 0.6465 0.6112 u24 1.1657 1.30461.3549 1.4052 u25 0.9479 1.0833 1.1931 1.2281 u26 0.8419 0.9785 1.01171.0054 u27 0.9302 1.0134 0.9857 0.9482 u28 0.9950 0.9901 0.9689 0.9425u29 0.8713 0.9610 0.9967 0.9939 u30 1.0185 1.0658 1.1683 1.1882 u311.4660 1.6424 1.6391 1.6566

Meanwhile, it needs to be noted that the method for obtaining completeset of constellation points as in Tables 10 and 11 is completely thesame as the method described in Tables 5 and 6.

Meanwhile, those skilled in the art may recognize that rotation, scaling(here, the scaling factor applied to a real axis and an imaginary axiscan be the same or different) or other transformation can be appliedwith respect to the constellation described above. The constellationindicates a comparative position of constellation points, and otherconstellation can be deduced through rotation, scaling, or othertransformation.

In addition, those skilled in the art can recognize that the inventiveconcept is not limited to constellation defined in Tables 2-11 describedabove.

For example, in certain exemplary embodiments, a constellation havingdifferent order and/or a constellation including a different arrangementor a comparative position of constellation points can be used. Asanother example, a constellation which is similar to one ofconstellations defined in Tables 2-11 can be used.

For example, a constellation which has values of constellation pointswith differences which do not exceed a predetermined threshold (orerror) from the values indicated in Tables 2-11 can be used. Here, thethreshold value can be expressed as comparative numbers (for example,0.1%, 1%, 5%, etc.), absolute numbers (for example, 0.001, 0.01, 0.1,etc.) or appropriate methods (rounding-off, flooring, ceiling, or thelike). As an example of rounding-off, constellation point“0.707316+0.707473i” can be approximated to “0.7073+0.7075i” byrounding-off at the five decimal places.

In addition, a transmitter and a receiver may use differentconstellations. For example, a transmitter and a receiver may userespective constellations which have at least one constellation pointthat has a difference which does not exceed a predetermined thresholdvalue. For example, a receiver may use a constellation having at leastone round off/round down constellation point (for example, A2) to demapconstellation points, whereas a transmitter may use a constellationhaving non-round off/round-down constellation points (for example, A1).

In addition, even if an order of the values in Tables 2-11 is changed,the set of constellation points itself is not changed, and thus, it ispossible to arrange the values by changing the order of values as inTables 2-11.

Hereinbelow, an example of a normalization method and an exemplaryembodiment of constituting 2D constellation from a 1D level set will bedescribed.

For example, in Table 12, it is assumed that values of constellationpoints of a 1D NU 1K QAM constellation for a 13/15 coding rate are asshown below.

TABLE 12 Coding Rate Level 13/15 1 1 2 2.975413 3 4.997551 4 7.018692 59.102872 6 11.22209 7 13.42392 8 15.69921 9 18.09371 10 20.61366 1123.2898 12 26.15568 13 29.23992 14 32.59361 15 36.30895 16 40.58404

Here, when a level vector A is indicated as A=(a_(i)), (I=0, 1, 2, . . ., L−1), first of all, the vector A is normalized using Equation 3 shownbelow, and normalized vector Ā can be obtained.

$\begin{matrix}{\overset{\_}{A} = \frac{A}{\sqrt{\frac{2}{L}{\sum\limits_{i}^{\;}\; a_{i}^{2}}}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

In above Equation 3, L indicates the number of level (that is,dimensionality of A). For example, in the case of 16-QAM, 64-QAM,256-QAM, 1024-QAM, and 4096-QAM, dimensionality of level can be 4, 6, 8,10 and 12 respectively.

In the example described above, the normalized vector A can be indicatedas Table 13 shown below.

TABLE 13 Coding Rate Level 13/15 1 0.0325 2 0.0967 3 0.1623 4 0.228 50.2957 6 0.3645 7 0.4361 8 0.51 9 0.5878 10 0.6696 11 0.7566 12 0.849713 0.9498 14 1.0588 15 1.1795 16 1.3184

If the normalization method described above is applied to Tables 5 and 6respectively, it can be easily recognized that Tables 10 and 11 will beobtained respectively.

Next, a final constellation is generated such that all the possiblecombinations of the real part and the imaginary part, which are the sameas one of the entries (that is, components). In this case, for anexample, gray mapping can be used.

In the example described above, constellation points in the final firstquadrant can be indicated as in Table 14 shown below.

TABLE 14 Label (int.) Constellation Point 1 1.3184 + 1.3184i 2 1.3184 +1.1795i 3 1.1795 + 1.3184i 4 1.1795 + 1.1795i 5 1.3184 + 0.9498i 61.3184 + 1.0588i 7 1.1795 + 0.9498i 8 1.1795 + 1.0588i 9 0.9498 +1.3184i 10 0.9498 + 1.1795i 11 1.0588 + 1.3184i 12 1.0588 + 1.1795i 130.9498 + 0.9498i 14 0.9498 + 1.0588i 15 1.0588 + 0.9498i 16 1.0588 +1.0588i 17 1.3184 + 0.5878i 18 1.3184 + 0.6696i 19 1.1795 + 0.5878i 201.1795 + 0.6696i 21 1.3184 + 0.8497i 22 1.3184 + 0.7566i 23 1.1795 +0.8497i 24 1.1795 + 0.7566i 25 0.9498 + 0.5878i 26 0.9498 + 0.6696i 271.0588 + 0.5878i 28 1.0588 + 0.6696i 29 0.9498 + 0.8497i 30 0.9498 +0.7566i 31 1.0588 + 0.8497i 32 1.0588 + 0.7566i 33 0.5878 + 1.1795i 340.6696 + 1.3184i 35 0.6696 + 1.1795i 36 0.5878 + 0.9498i 37 0.5878 +1.0588i 38 0.6696 + 0.9498i 39 0.6696 + 1.0588i 40 0.8497 + 1.3184i 410.8497 + 1.1795i 42 0.7566 + 1.3184i 43 0.7566 + 1.1795i 44 0.8497 +0.9498i 45 0.8497 + 1.0588i 46 0.7566 + 0.9498i 47 0.7566 + 1.0588i 480.5878 + 0.5878i 49 0.5878 + 0.6696i 50 0.6696 + 0.5878i 51 0.6696 +0.6696i 52 0.5878 + 0.8497i 53 0.5878 + 0.7566i 54 0.6696 + 0.8497i 550.6696 + 0.7566i 56 0.8497 + 0.5878i 57 0.8497 + 0.6696i 58 0.7566 +0.5878i 59 0.7566 + 0.6696i 60 0.8497 + 0.8497i 61 0.8497 + 0.7566i 620.7566 + 0.8497i 63 0.7566 + 0.7566i 64 1.3184 + 0.0325i 65 1.3184 +0.0967i 66 1.1795 + 0.0325i 67 1.1795 + 0.0967i 68 1.3184 + 0.2280i 691.3184 + 0.1623i 70 1.1795 + 0.2280i 71 1.1795 + 0.1623i 72 0.9498 +0.0325i 73 0.9498 + 0.0967i 74 1.0588 + 0.0325i 75 1.0588 + 0.0967i 760.9498 + 0.2280i 77 0.9498 + 0.1623i 78 1.0588 + 0.2280i 79 1.0588 +0.1623i 80 1.3184 + 0.5100i 81 1.3184 + 0.4361i 82 1.1795 + 0.5100i 831.1795 + 0.4361i 84 1.3184 + 0.2957i 85 1.3184 + 0.3645i 86 1.1795 +0.2957i 87 1.1795 + 0.3645i 88 0.9498 + 0.5100i 89 0.9498 + 0.4361i 901.0588 + 0.5100i 91 1.0588 + 0.4361i 92 0.9498 + 0.2957i 93 0.9498 +0.3645i 94 1.0588 + 0.2957i 95 1.0588 + 0.3645i 96 0.5878 + 0.0325i 970.5878 + 0.0967i 98 0.6696 + 0.0325i 99 0.6696 + 0.0967i 100 0.5878 +0.2280i 101 0.5878 + 0.1623i 102 0.6696 + 0.2280i 103 0.6696 + 0.1623i104 0.8497 + 0.0325i 105 0.8497 + 0.0967i 106 0.7566 + 0.0325i 1070.7566 + 0.0967i 108 0.8497 + 0.2280i 109 0.8497 + 0.1623i 110 0.7566 +0.2280i 111 0.7566 + 0.1623i 112 0.5878 + 0.5100i 113 0.5878 + 0.4361i114 0.6696 + 0.5100i 115 0.6696 + 0.4361i 116 0.5878 + 0.2957i 1170.5878 + 0.3645i 118 0.6696 + 0.2957i 119 0.6696 + 0.3645i 120 0.8497 +0.5100i 121 0.8497 + 0.4361i 122 0.7566 + 0.5100i 123 0.7566 + 0.4361i124 0.8497 + 0.2957i 125 0.8497 + 0.3645i 126 0.7566 + 0.2957i 1270.7566 + 0.3645i 128 0.0325 + 1.3184i 129 0.0325 + 1.1795i 130 0.0967 +1.3184i 131 0.0967 + 1.1795i 132 0.0325 + 0.9498i 133 0.0325 + 1.0588i134 0.0967 + 0.9498i 135 0.0967 + 1.0588i 136 0.2280 + 1.3184i 1370.2280 + 1.1795i 138 0.1623 + 1.3184i 139 0.1623 + 1.1795i 140 0.2280 +0.9498i 141 0.2280 + 1.0588i 142 0.1623 + 0.9498i 143 0.1623 + 1.0588i144 0.0325 + 0.5878i 145 0.0325 + 0.6696i 146 0.0967 + 0.5878i 1470.0967 + 0.6696i 148 0.0325 + 0.8497i 149 0.0325 + 0.7566i 150 0.0967 +0.8497i 151 0.0967 + 0.7566i 152 0.2280 + 0.5878i 153 0.2280 + 0.6696i154 0.1623 + 0.5878i 155 0.1623 + 0.6696i 156 0.2280 + 0.8497i 1570.2280 + 0.7566i 158 0.1623 + 0.8497i 159 0.1623 + 0.7566i 160 0.5100 +1.3184i 161 0.5100 + 1.1795i 162 0.4361 + 1.3184i 163 0.4361 + 1.1795i164 0.5100 + 0.9498i 165 0.5100 + 1.0588i 166 0.4361 + 0.9498i 1670.4361 + 1.0588i 168 0.2957 + 1.3184i 169 0.2957 + 1.1795i 170 0.3645 +1.3184i 171 0.3645 + 1.1795i 172 0.2957 + 0.9498i 173 0.2957 + 1.0588i174 0.3645 + 0.9498i 175 0.3645 + 1.0588i 176 0.5100 + 0.5878i 1770.5100 + 0.6696i 178 0.4361 + 0.5878i 179 0.4361 + 0.6696i 180 0.5100 +0.8497i 181 0.5100 + 0.7566i 182 0.4361 + 0.8497i 183 0.4361 + 0.7566i184 0.2957 + 0.5878i 185 0.2957 + 0.6696i 186 0.3645 + 0.5878i 1870.3645 + 0.6696i 188 0.2957 + 0.8497i 189 0.2957 + 0.7566i 190 0.3645 +0.8497i 191 0.3645 + 0.7566i 192 0.0325 + 0.0325i 193 0.0325 + 0.0967i194 0.0967 + 0.0325i 195 0.0967 + 0.0967i 196 0.0325 + 0.2280i 1970.0325 + 0.1623i 198 0.0967 + 0.2280i 199 0.0967 + 0.1623i 200 0.2280 +0.0325i 201 0.2280 + 0.0967i 202 0.1623 + 0.0325i 203 0.1623 + 0.0967i204 0.2280 + 0.2280i 205 0.2280 + 0.1623i 206 0.1623 + 0.2280i 2070.1623 + 0.1623i 208 0.0325 + 0.5100i 209 0.0325 + 0.4361i 210 0.0967 +0.5100i 211 0.0967 + 0.4361i 212 0.0325 + 0.2957i 213 0.0325 + 0.3645i214 0.0967 + 0.2957i 215 0.0967 + 0.3645i 216 0.2280 + 0.5100i 2170.2280 + 0.4361i 218 0.1623 + 0.5100i 219 0.1623 + 0.4361i 220 0.2280 +0.2957i 221 0.2280 + 0.3645i 222 0.1623 + 0.2957i 223 0.1623 + 0.3645i224 0.5100 + 0.0325i 225 0.5100 + 0.0967i 226 0.4361 + 0.0325i 2270.4361 + 0.0967i 228 0.5100 + 0.2280i 229 0.5100 + 0.1623i 230 0.4361 +0.2280i 231 0.4361 + 0.1623i 232 0.2957 + 0.0325i 233 0.2957 + 0.0967i234 0.3645 + 0.0325i 235 0.3645 + 0.0967i 236 0.2957 + 0.2280i 2370.2957 + 0.1623i 238 0.3645 + 0.2280i 239 0.3645 + 0.1623i 240 0.5100 +0.5100i 241 0.5100 + 0.4361i 242 0.4361 + 0.5100i 243 0.4361 + 0.4361i244 0.5100 + 0.2957i 245 0.5100 + 0.3645i 246 0.4361 + 0.2957i 2470.4361 + 0.3645i 248 0.2957 + 0.5100i 249 0.2957 + 0.4361i 250 0.3645 +0.5100i 251 0.3645 + 0.4361i 252 0.2957 + 0.2957i 253 0.2957 + 0.3645i254 0.3645 + 0.2957i 255 0.3645 + 0.3645i 256 1.3184 − 1.3184i

FIG. 35 is a block diagram to describe a configuration of a transmittingapparatus according to an exemplary embodiment. Referring to FIG. 35,the transmitting apparatus 3500 includes an encoder 3510, an interleaver3520, and a modulator 3530 (or, ‘constellation mapper’).

The encoder 3510 performs channel encoding with respect to input bitsand generates a codeword.

For example, the encoder 3510 may perform LDPC encoding with respect tothe bits and generate an LDPC codeword using an LDPC encoder (notshown).

Specifically, the encoder 3510 may perform LDPC encoding with the inputbits as the information word bits, and generate the LDPC codewordconstituting the information word bits and parity bits (that is, theLDPC parity bits). In this case, the LDPC code is a systematic code, theinformation word may be included in the LDPC codeword as it is.

Herein, the LDPC codeword is constituted by the information word bitsand parity bits. For example, the LDPC codeword has N_(ldpc) bits, andmay include the information word bits formed of K_(ldpc) bits and paritybits formed of N_(parity)=N_(ldpc)−K_(ldpc) parity bits.

In this case, the encoder 3510 may perform LDPC encoding based on aparity check matrix and generate the LDPC codeword. That is, a processof performing LDPC encoding is a process of generating the LDPC codewordsatisfying H·C^(T)=0, and thus, the encoder 3510 may use the paritycheck matrix when performing LDPC encoding. Herein, H is a parity checkmatrix and C is an LDPC codeword.

To do this, the transmitting apparatus 3500 may include a separatememory and prestore various types of a parity check matrix.

However, this is merely exemplary, and channel encoding may be performedin various schemes.

The encoder 3510 may perform channel encoding using various coding ratessuch as 2/15, 3/15, 4/15, 5/15, 6/15, 7/15, 8/15, 9/15, 10/15, 11/15,12/15 and 13/15. In addition, the encoder 3510 may generate a codewordhaving various lengths such as 16200 and 64800 based on a length of thebits and coding rate.

An interleaver 3520 interleaves the codeword. That is, the interleaver3520, based on various interleaving rules, may perform bit-interleavingof the codeword generated by the encoder 3510.

A modulator 3530 maps the codeword which is interleaved according to amodulation scheme onto a non-uniform constellation.

Specifically, the modulator 3530 may perform serial-to-parallelconversion with respect to the interleaved codeword, and demultiplex theinterleaved codeword into a cell (or a cell word) formed of a certainnumber of bits.

For example, the modulator 3530 may receive the codeword bitsQ=(q₀,q₁,q₂, . . . ) output from the interleaver 3520, and generatescells.

In this case, the number of bits constituting each cell may be the sameas the number of bits constituting a modulation symbol (that is, amodulation order). For example, when the modulator 3530 performsmodulation using QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM, 4096-QAM, thenumber of bits η_(MOD) constituting the modulation symbol may be 2, 4,6, 8, 10 and 12.

For example, when the modulation scheme is 64-QAM, η_(MOD) is 6(η_(MOD)=6), and thus, each cell may be composed as (q₀,q₁,q₂,q₃,q₄,q₅),(q₆,q₇,q₈,q₉,q₁₀,q₁₁), (q₁₂,q₁₃,q₁₄,q₁₅,q₁₆,q₁₇), . . . .

In addition, the modulator 3530 may perform modulation by mapping thecells onto the non-uniform constellation.

Specifically, each cell includes bits as many as the number constitutingthe modulation symbol, and thus, the modulator 3530 may generate themodulation symbol by sequentially mapping each cell onto a constellationpoint of the non-uniform constellation. Herein, the modulation symbolcorresponds to a constellation point of a constellation.

In this case, constellation may include constellation points which aredefined based on Tables 2-11 according to a modulation scheme.

To be specific, the constellation may include the constellation pointswhich are defined by a constellation position vector as in Tables 2-4and 7-9, according to a modulation scheme. Or, the constellation mayinclude the constellation points which are defined by the constellationposition vector which is generated based on the level set as in Tables5, 6, 10, and 11 according to a modulation scheme.

That is, the modulator 3530, in consideration of the coding rate usedfor encoding by the encoder 3510, may perform modulation by mappingcells onto the set of constellation points which corresponds to thecoding rate from among the sets of constellation points which aredefined based on Tables 2-11 according to the coding rates.

For example, constellation may include constellation points which aredefined based on Table 8, when a modulation scheme is 64-QAM.

To be specific, the modulator 3530, when encoding is performed with thecoding rate of 2/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes constellationpoints defined by NUC_64_2/15 of Table 8.

That is, when the coding rate is 2/15 and modulation is performed to 2D64NUC, the constellation points in the first quadrant of constellationcan be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₁₄,w₁₅}={0.6474+0.9831i, 0.6438+0.9829i, 0.6471+0.9767i, . . . ,0.6385+0.4656i, 0.6353+0.4653i} which is defined as NUC_64_2/15 of Table8.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 3/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_64_3/15 of Table 8.

That is, when the coding rate is 3/14 and modulation is performed to 2D64NUC, the constellation points in the first quadrant of constellationcan be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₁₄,w₁₅}={0.5472+1.1591i, 0.5473+1.1573i, 0.5467+1.1599i, . . . ,0.5087+0.3163i, 0.5087+0.3163i} which is defined as NUC_64_3/15 of Table8.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 4/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_64_4/15 of Table 8.

That is, when coding rate is 4/15 and modulation is performed to 2D64NUC, the constellation points of the first quadrant of constellationmay be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₁₄,w₁₅}={0.5008+1.2136i, 0.4994+1.2194i, 0.5313+1.1715i, . . . ,0.4791+0.2773i, 0.4791+0.2758i} which is defined as NUC_64_4/15 of Table8.

Table 8 indicates the constellation points in one quadrant ofconstellation, and the constellation points in remaining quadrants ofconstellation may be obtained by indicating each constellation point a,which is defined in Table 8, as a*, −a* and −a respectively. (Here, *indicates complex conjugation).

As another example, when the modulation scheme is 256-QAM, theconstellation points which are defined based on Table 9 may be included.

Specifically, the modulator 3530, when encoding is performed with thecoding rate of 2/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_256_2/15 of Table 9.

That is, when coding rate is 2/15 and modulation is performed to 2D256NUC, the constellation points of the first quadrant of constellationmay be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₆₂,w₆₃}={0.5553+1.1262i, 0.5673+0.1336i, 0.5593+1.1204i, . . . ,0.5319+0.3381i, 0.5327+0.3395i} which is defined as NUC_256_2/15 ofTable 9.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 3/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_256_3/15 of Table 9.

That is, when coding rate is 3/15 and modulation is performed to 2D256NUC, the constellation points of the first quadrant of constellationmay be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₆₂,w₆₃}={0.5229+1.1810i, 0.5384+1.1625i, 0.5148+1.1943i, . . . ,0.4734+0.2696i, 0.4749+0.2711i} which is defined as NUC_256_3/15 ofTable 9.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 4/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_256_4/15 of Table 8.

That is, when coding rate is 4/15 and modulation is performed to 2D256NUC, the constellation points of the first quadrant of constellationmay be expressed as the constellation position vector {w₀,w₁,w₂, . . .,w₆₂,w₆₃}={0.2975+1.0564i, 0.5862+0.9617i, 0.2909+1.0696i, . . . ,0.3762+0.1998i, 0.3689+0.2114i} which is defined as NUC_256_4/15 ofTable 9.

Table 9 indicates the constellation points in one quadrant ofconstellation, and the constellation points in remaining quadrants ofconstellation may be obtained by indicating each constellation point a,which is defined in Table 9, as a*, −a* and −a respectively (Here, *indicates complex conjugation).

As another example, constellation, when the modulation scheme is1024-QAM, may include the constellation points which are defined basedon Table 10.

Specifically, the modulator 3530, when encoding is performed with thecoding rate of 2/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_1k_2/15 of Table 10.

That is, when coding rate is 2/15 and modulation is performed to 1D1024NUC, the level set may be A={0.3317, 0.3321, 0.3322, . . . , 0.9394,0.9349} as NUC_1k_2/15 of Table 10, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.3317+0.3317i, 0.3317+0.3321i, 0.3321+0.3317i, . . . ,0.9349+0.9349i}.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 3/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_1k_3/15 of Table 10.

That is, when coding rate is 3/15 and modulation is performed to 1D1024NUC, level set may be A={0.2382, 0.2556, 0.2749, . . . , 0.9459,1.4299} as NUC_1k_3/15 of Table 10, and the constellation positionvector indicating the constellation points of the first quadrant may beexpressed as {0.2382+0.2382i, 0.2382+0.2556i, 0.2556+0.2382i, . . . ,1.4299+1.4299i}.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 4/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_1k_4/15 of Table 10.

That is, when coding rate is 4/15 and modulation is performed to 1D1024NUC, the level set may be A={0.1924, 0.1940, 0.2070, . . . , 1.1332,1.4761} as NUC_1k_4/15 of Table 10, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.1924+0.1924i, 0.1924+0.1940i, 0.1940+0.1924i, . . . ,1.4761+1.4761i}.

Table 10 is used to define the constellation points in one quadrant ofconstellation, and the constellation points in remaining quadrants maybe obtained by indicating each constellation point, which is definedbased on Table 10, as a*, −a* and −a (Here, * indicates complexconjugation).

As another example, constellation, when the modulation scheme is4096-QAM, may include the constellation points which are defined basedon Table 11.

Specifically, the modulator 3530, when encoding is performed with thecoding rate 2/15 by the encoder 3510, may map the interleaved codewordonto the non-uniform constellation which includes the constellationpoints defined by NUC_4k_2/15 of Table 11.

That is, when coding rate is 2/15 and modulation is performed to 1D4096NUC, the level set may be A={0.2826, 0.2885, 0.2944, . . . , 1.0185,1.4660} as NUC_4k_2/15 of Table 11, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.2826+0.2826i, 0.2826+0.2885i, 0.2885+0.2826i, . . . ,1.4660+1.4660i}.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 3/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_4k_3/15 of Table 11.

That is, when coding rate is 3/15 and modulation is performed to 1D4096NUC, level set may be A={0.2038, 0.2038, 0.2155, . . . , 1.0658,1.6424} as NUC_4k_3/15 of Table 11, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.2038+0.2038i, 0.2038+0.2155i, 0.2155+0.2038i, . . . ,1.6424+1.6424i}.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 4/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_4k_4/15 of Table 11.

That is, when coding rate is 4/15 and modulation is performed to 1D4096NUC, the level set may be A={0.1508, 0.1468, 0.1456, . . . , 1.1683,1.6391} as NUC_4k_4/15 of Table 11, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.1508+0.1508i, 0.1508+0.1468i, 0.1468+0.1508i, . . . ,1.6391+1.6391i}.

In addition, the modulator 3530, when encoding is performed with thecoding rate of 4/15 by the encoder 3510, may map the interleavedcodeword onto the non-uniform constellation which includes theconstellation points defined by NUC_4k_5/15 of Table 11.

That is, when coding rate is 5/15 and modulation is performed to 1D4096NUC, the level set may be A={0.1257, 0.1257, 0.1257, . . . , 1.1882,1.6566} as NUC_4k_5/15 of Table 11, and the constellation positionvector indicating the constellation points in the first quadrant may beexpressed as {0.1257+0.1257i, 0.1257+0.3599i, 0.3599+0.1257i, . . . ,1.6566+1.6566i}.

Table 11 is used to define the constellation points in one quadrant, andthe constellation points with respect to the remaining quadrants ofconstellation may be obtained by indicating each constellation point a,which is defined based on Table 11, as a*, −a* and −a (Here, * indicatescomplex conjugation).

In the above-described examples, it is described that the cells aremapped onto the set of constellation points which correspond to codingrate used for encoding, but this is merely exemplary, and in some cases,the modulator 3530 may map the cells onto the set of constellationpoints which do not correspond to coding rate which is used forencoding.

As an example, when 64-QAM is used, even if encoding is performed withthe coding rate of 2/15, the modulator 3530 may map the cells onto theset of constellation points which are defined as NUC_64_3/15 orNUC_64_4/15 of Table 8, instead of the set of constellation points whichare defined as NUC_64_2/15 of Table 8.

The transmitting apparatus 3500 may modulate a signal which is mappedonto the constellation and transmit the signal to a receiving apparatus(for example, 3600 of FIG. 36). For example the transmitting apparatus3500 may map the signal which is mapped to the constellation onto anorthogonal frequency division multiplexing (OFDM) frame by using an OFDMscheme, and may transmit the signal to the receiving apparatus 3600 viaan allocated channel.

In other word, in mapping method for 16-QAM, 64-QAM and 256-QAM, eachinput data cell word (y_(0,s), . . . , y_(ηMOD-1,s)) shall be modulatedusing a 2D non-uniform constellation to give a constellation pointz_(s). Index s denotes the discrete time index, η_(MOD)=log₂(M), M beingthe number of constellation points, e.g., M=64 for 64-QAM. The vector ofcomplex constellation points x=(x₀, . . . , x_(M-1)) includes all Mconstellation points of the QAM alphabet. The k-th element of thisvector, x_(k), corresponds to the QAM constellation point for the inputcell word (y_(0,s), . . . , y_(ηMOD-1,s)), if these bits take on thedecimal number k (y_(0,s) being the most significant bit (MSB), andy_(ηMOD-1,s) being the least significant bit (LSB)). Due to the quadrantsymmetry, the complete vector x can be derived by defining just thefirst quarter of the complex constellation points, i.e., (x₀, . . . ,x_(M/4-1)), which corresponds to the first quadrant. The generation rulefor the remaining points is described below. Defining b=M/4, the firstquarter of complex constellation points is denoted as the NUC positionvector w=(w₀, . . . , w_(b-1)). The position vectors are defined theabove tables. As an example, the NUC position vector for a 16-QAMcomprises the complex constellation points with the labels correspondingto the decimal values 0, i.e., (y_(0,s), y_(ηMOD-1,s))=0000, to b−1,i.e., (y_(0,s), . . . , y_(ηMOD-1,s))=0011. The remaining constellationpoints are derived as follows:

(x₀, . . . , x_(b-1))=w (first quarter)

(x_(b), . . . , x_(2b-1))=−conj(w) (second quarter)

(x_(2b), . . . , x_(3b-1))=conj(w) (third quarter)

(x_(3b), . . . , x_(4b-1))=−w (fourth quarter),

with conj being the complex conjugate.

As an example, the NUC position vector for 16-QAM and code rate 2/15 isconstructed as follows. From Table 7, w=(0.7073+0.7075i, 0.7073+0.7074i,0.7060+0.7077i, 0.7065+0.7071i). Here and in the following, i=√(−1) isthe imaginary unit. Assuming the input data cell word is (y_(0,s), . . ., y_(ηMOD-1,s))=(1100), the corresponding QAM constellation point attime index s is z_(s)=x₁₂=−w₀=−0.7073−0.7075i.

Also, in mapping method for 1024-QAM and 4096-QAM, Each input data cellword (y_(0,s), . . . , y_(ηMOD-1,s)) at discrete time index s shall bemodulated using a 1-dimensional non-uniform QAM constellation to give aconstellation point z_(s) prior to normalization. 1-dimensional refersto the fact that a 2-dimensional QAM constellation can be separated intotwo 1-dimensional PAM constellations, one for each I and Q component.The exact values of the real and imaginary components Re(z_(s)) andIm(z_(s)) for each combination of the relevant input cell word (y_(0,s),. . . , y_(ηMOD-1,s)) are given by a 1D-NUC position vector u=(u₀, . . .u_(v)), which defines the constellation point positions of thenon-uniform constellation in one dimension. The number of elements ofthe 1D-NUC position vector u is defined by

$v = {\frac{\sqrt{M}}{2}.}$

As an example the 1024-NUC for code rate 2/15 is defined by the NUCposition vector NUC_1k_2/15. From Table 10, u=(u₀, . . . , u₁₅)=(0.3317,0.3321, 0.3322, 0.3321, 0.3327, 0.3328, 0.3322, 0.3322, 0.9369, 0.9418,0.9514, 0.9471, 0.9448, 0.9492, 0.9394, 0.9349). Assuming the input datacell (y_(0,s), . . . , y_(ηMOD-1,s))=(0010011100) the corresponding QAMconstellation point z_(s) has Re(z_(s))=u₃=0.3321 (defined by even indexbit labels, i.e., 01010) and Im(z_(s))=u₁₁=0.9471 (defined by odd indexbit label, i.e., 00110).

FIG. 36 is a block diagram to describe a configuration of the receivingapparatus according to an exemplary embodiment. Referring to FIG. FIG.36, the receiving apparatus 3600 includes a demodulator 3610, adeinterleaver 3620, and a decoder 3630.

The demodulator 3610 receives and demodulates a signal transmitted fromthe transmitting apparatus 3500. Specifically, the demodulator 3610 maygenerate a value corresponding to the codeword by demodulating thereceived signal.

In this case, the demodulator 3610 may perform demodulation tocorrespond to the modulation scheme which is used by the transmittingapparatus 3500. To do this, the transmitting apparatus 3500 may transmitinformation on the modulation scheme to the receiving apparatus 3600, orthe transmitting apparatus 3500 may perform modulation using themodulation scheme which is predefined between the transmitting apparatus3500 and the receiving apparatus 3600.

Meanwhile, a value which corresponds to the codeword may be expressed asa channel value with respect to the received signal. There may bevarious methods for determining the channel value, for example, a methodfor determining a log likelihood ratio (LLR) value is an example of themethod for determining the channel value.

The LLR value may indicate a log value for a ratio of the probabilitythat the bit transmitted from the transmitting apparatus 3500 is 0 andthe probability that the bit is 1. In addition, the LLR value may be abit value which is determined by a hard decision, or may be arepresentative value which is determined according to a section to whichthe probability that the bit transmitted from the transmitting apparatus3500 is 0 or 1 belongs.

The demodulator 3610 may perform cell-to-bit conversion with respect toa value corresponding to the codeword and output an LLR value in theunit of bits.

The deinterleaver 3620 deinterleaves an output value of the demodulator3610, and outputs the value to the decoder 3630.

To be specific, the deinterleaver 3620 is an element corresponding tothe interleaver 3520 of the transmitting apparatus 3500 and performs anoperation corresponding to the interleaver 3520. That is, theinterleaver 3620 performs the interleaving operation of the interleaver3520 inversely and deinterleaves an LLR value.

The decoder 3630 may perform channel decoding based on the output valueof the deinterleaver 3620.

Specifically, the decoder 3630 is an element corresponding to theencoder 3510 of the transmitting apparatus 3500, which may correct anerror by performing decoding by using the LLR value output from thedeinterleaver 3620.

For example, the decoder 3630 may include an LDPC decoder (not shown) toperform LDPC decoding.

In this case, the decoder 3630 may perform LDPC decoding using aniterative decoding scheme based on a sum-product algorithm. Herein, thesum-product algorithm refers to an algorithm by which messages (e.g.,LLR value) are exchanged through an edge on a bipartite graph of amessage passing algorithm, and an output message is calculated frommessages input to variable nodes or check nodes, and is updated.

Meanwhile, the decoder 3630 may use a parity check matrix for LDPCdecoding. In this case, the parity check matrix which is used fordecoding may have the same structure as the parity check matrix which isused for encoding.

Meanwhile, information on the parity check matrix or information on thecode rate used for LDPC decoding may be prestored in the receivingapparatus 3600 or provided by the transmitting apparatus 3500.

The foregoing is merely exemplary, and channel decoding may be performedby various schemes which correspond to the channel coding which isperformed by the transmitting apparatus 3500.

A non-transitory computer readable medium may be provided which stores aprogram to operate the above-described methods of various exemplaryembodiments, according to an exemplary embodiment. The non-transitoryrecordable medium refers to a medium which may store datasemi-permanently rather than storing data for a short time such as aregister, a cache, and a memory and may be readable by an apparatus.Specifically, this medium may be a non-temporal recordable medium suchas compact disk (CD), digital versatile disk (DVD), hard disk, Blu-raydisk, universal serial bus (USB), memory card, or read-only memory(ROM), not being limited thereto.

The components, elements, modules or units represented by a block asillustrated in FIGS. 18, 35 and 36 may be embodied as various numbers ofhardware, software and/or firmware structures that execute respectivefunctions described above, according to an exemplary embodiment. Forexample, these components, elements, modules or units may use a directcircuit structure, such as a memory, processing, logic, a look-up table,etc. that may execute the respective functions through controls of oneor more microprocessors or other control apparatuses. Also, thesecomponents, elements, modules or units may be specifically embodied by aprogram or a part of code, which contains one or more executableinstructions for performing specified logic functions. Also, at leastone of these components, elements, modules or units may further includea processor such as a central processing unit (CPU) that performs therespective functions, a microprocessor, or the like. A bus is notillustrated in the above block diagrams of FIGS. 18, 35 and 36. However,communications between the respective components, elements, modules orunits may be carried out through the bus.

The foregoing embodiments and advantages are merely exemplary and shouldnot be construed as limiting the inventive concept. Also, thedescription of the exemplary embodiments of the inventive concept isintended to be illustrative, and not to limit the scope of the claims,and many alternatives, modifications, and variations will be apparent tothose skilled in the art.

FIG. 37 is a flowchart to describe a method for transmitting of atransmitting apparatus according to an exemplary embodiment.

First of all, a codeword is generated (S3710) by performing channelencoding with respect to the bits, and the codeword is interleaved(S3720).

Thereafter, the interleaved codeword is mapped onto the non-uniformconstellation according to a modulation scheme (S3730).

In this case, constellation may include the constellation points whichare defined based on Tables 2-11 according to a modulation scheme.

As an example, when the modulation scheme is 64-QAM, constellation mayinclude the constellation points which are defined based on Table 8.

Specifically, when encoding is performed with the coding rate of 2/15,at S3730, an interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined byNUC_64_2/15 of Table 8.

When encoding is performed with the coding rate of 3/15, at S3730, theinterleaved codeword may be mapped onto the non-uniform constellationwhich includes the constellation points which are defined by NUC_64_3/15of Table 8.

In addition, when encoding is performed with the coding rate of 4/15, atS3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined byNUC_64_4/15 of Table 8.

As another example, constellation may include the constellation pointswhich are defined based on Table 9, when the modulation scheme is256-QAM.

Specifically, when encoding is performed with the coding rate of 2/15,at S3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined byNUC_256_2/15 of Table 9.

In addition, when encoding is performed with the coding rate of 3/15, atS3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points which are definedby NUC_256_3/15 of Table 9.

When encoding is performed with the coding rate of 4/15, at S3730, theinterleaved codeword may be mapped onto the non-uniform constellationwhich includes the constellation points defined by NUC_256_4/15 of Table9.

Meanwhile, Tables 9 and 9 indicate the constellation points in onequadrant of the constellation, and the constellation points in theremaining quadrants of constellation may be obtained by indicating eachconstellation point a, which is defined in Tables 8 and 9, as a*, −and−a respectively (Here, * indicates complex conjugation).

As another example, constellation may include, when the modulationscheme is 1024-QAM, the constellation points which are defined based onTable 10.

Specifically, when encoding is performed with the coding rate of 2/15,at S3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined based onNUC_1k_2/15 of Table 10.

In addition, when encoding is performed with the coding rate of 3/15, atS3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined based onNUC_1k_3/15 of Table 10.

In addition, when encoding is performed with the coding rate of 4/15, atS3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined based onNUC_1k_4/15 of Table 10.

As another example, constellation may include the constellation pointswhich are defined based on Table 11, when the modulation scheme is4096-QAM.

Specifically, when encoding is performed with the coding rate of 2/15,at S3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined based onNUC_4k_2/15 of Table 11.

In addition, when encoding is performed with the coding rate of 3/15, atS3730, the interleaved codeword may be mapped onto the non-uniformconstellation which includes the constellation points defined based onNUC_4k_3/15 of Table 11.

When encoding is performed with the coding rate of 4/15, at S3730, theinterleaved codeword may be mapped onto the non-uniform constellationwhich includes the constellation points which are defined based onNUC_4k_4/15 of Table 11.

When encoding is performed with the coding rate of 5/15, at S3730, theinterleaved codeword may be mapped onto the non-uniform constellationwhich includes the constellation points which are defined based onNUC_4k_5/15 of Table 11.

Meanwhile, Tables 10 and 11 are used to define the constellation pointsin one quadrant, and the constellations in the remaining quadrants ofconstellation may be obtained by indicating each constellation point a,which is defined based on Tables 10 and 11, as a*, −a* and −arespectively (Here, * indicates complex conjugation).

In the present disclosure, in order to generate the optimizedconstellation, capacity needs to be determined. To do this, SNR is animportant parameter. However, optimizing with respect to the SNR doesnot necessarily mean that an environment which satisfies the SNR isnecessary. Though it is highly likely that the optimized performance maybe obtained in the environment which satisfies the SNR, but in general,receiving SNR may change frequently according to system environment, andit is obvious that different SNR or different channel coding rate may beused according to not only complexity of realizing the system but alsovarious purposes to support several channel environments using themodulation scheme to which one NUC constellation point is applied.

A non-transitory computer readable medium in which a program whichsequentially performs non-uniform constellation generation method isstored therein may be provided.

The non-transitory computer-recordable medium is not a medium configuredto temporarily store data such as a register, a cache, or a memory butan apparatus-readable medium configured to semi-permanently store data.Specifically, the above-described various applications or programs maybe stored in the non-transitory apparatus-readable medium such as acompact disc (CD), a digital versatile disc (DVD), a hard disc, aBlu-ray disc, a universal serial bus (USB), a memory card, or a readonly memory (ROM), and provided.

In the block diagram which illustrates the transmitting apparatus andreceiving apparatus, bus is nogt illustrated, but communication amongelements of each apparatus can be done through the bus. In addition,each apparatus may further include CPU performing the steps describedabove and processors such as a micro processor.

The foregoing exemplary embodiments and advantages are merely exemplaryand are not to be construed as limiting the inventive concept. Theexemplary embodiments can be readily applied to other types of device orapparatus. Also, the description of the exemplary embodiments isintended to be illustrative, and not to limit the scope of the inventiveconcept, and many alternatives, modifications, and variations will beapparent to those skilled in the art.

What is claimed is:
 1. A receiving apparatus comprising: a receiver configured to receive a signal from a transmitting apparatus; a demodulator configured to demodulate the signal to generate values; a deinterleaver configured to deinterleave the values; and a decoder configured to decode the deinterleaved values based on a low density parity check (LDPC) code, a code rate of the LDPC code being 3/15, wherein the signal is demodulated based on 64-quadrature amplitude modulation (QAM), position vectors w, in which w comprises w₀, w₁, . . . , and w₁₅ and a quadrant of the position vector w_(i) in which i is any one of 0 to 15, and wherein the position vectors w are represented in a table as below: w₀ 0.5472 + 1.1591i w₁ 0.5473 + 1.1573i w₂ 0.5467 + 1.1599i w₃ 0.5479 + 1.1585i w₄ 1.1578 + 0.5478i w₅ 1.1576 + 0.5475i w₆ 1.1591 + 0.5475i w₇ 1.1591 + 0.5475i w₈ 0.3163 + 0.5072i w₉ 0.3163 + 0.5072i w₁₀ 0.3163 + 0.5072i w₁₁ 0.3163 + 0.5072i w₁₂ 0.5087 + 0.3163i w₁₃ 0.5087 + 0.3163i w₁₄ 0.5087 + 0.3163i w₁₅  0.5087 + 0.3163i.


2. The receiving apparatus as claimed in claim 1, wherein constellation points for 64-QAM comprise constellation points in 4 quadrants, wherein the represented position vectors w indicate constellation points in one quadrant, and wherein constellation points in remaining quadrants are obtained by indicating each constellation point a which is defined by the table as a*, −a* and −a respectively, where * indicates complex conjugation.
 3. A transmitting apparatus comprising: an interleaver configured to interleave a codeword; and a mapper configured to demultiplex bits of the interleaved codeword to generate cells, and map the cells to 2-dimensional (2D) non-uniform constellation points using 64-quadrature amplitude modulation (QAM) and position vectors w, in which w comprises w₀, w₁, . . . , and w₁₅, wherein the codeword comprises input bits and parity bits which are generated by encoding the input bits based on a low density parity check (LDPC) code, a code rate of the LDPC code being 3/15, wherein the cells are mapped to the constellation points by determining a position vector w_(i), in which i is any one of 0 to 15, corresponding to a cell from among the position vectors w, and determining a quadrant of the position vector w_(i), and wherein the position vectors w are represented in a table as below: w₀ 0.5472 + 1.1591i w₁ 0.5473 + 1.1573i w₂ 0.5467 + 1.1599i w₃ 0.5479 + 1.1585i w₄ 1.1578 + 0.5478i w₅ 1.1576 + 0.5475i w₆ 1.1591 + 0.5475i w₇ 1.1591 + 0.5475i w₈ 0.3163 + 0.5072i w₉ 0.3163 + 0.5072i w₁₀ 0.3163 + 0.5072i w₁₁ 0.3163 + 0.5072i w₁₂ 0.5087 + 0.3163i w₁₃ 0.5087 + 0.3163i w₁₄ 0.5087 + 0.3163i w₁₅  0.5087 + 0.3163i.


4. The transmitting apparatus as claimed in claim 3, wherein constellation points for 64-QAM comprise constellation points in 4 quadrants, wherein the represented position vectors w indicate constellation points in one quadrant, and wherein constellation points in remaining quadrants are obtained by indicating each constellation point a which is defined by the table as a*, −a* and −a respectively, where * indicates complex conjugation. 